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Różniczkowanie macierzy

In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics.

Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is made that vectors should be treated as column vectors when combined with matrices (rather than row vectors). A single convention can be somewhat standard throughout a single field that commonly use matrix calculus (e.g. econometrics, statistics, estimation theory and machine learning). However, even within a given field different authors can be found using competing conventions. Authors of both groups often write as though their specific convention is standard. Serious mistakes can result when combining results from different authors without carefully verifying that compatible notations are used. Therefore great care should be taken to ensure notational consistency. Definitions of these two conventions and comparisons between them are collected in the layout conventions section.

Scope

Rachunek macierzowy korzysta z różnych notacji, które zbierają w macierze lub wektory pochodną każdej składowej zmiennej zależnej po każdej składowej zmiennej zależnej. W ogólności zmienną niezależną może być skalar, wektor lub macierz. Zmienną zależną może być także każdy z wyżej wymienionych. Zapis macierzy umożliwia poręczne zapis tych pochodnych.

Przykładowo rozważmy gradient wektora. Dla funkcji skalarnej trzech niezależnych zmiennych $f(x_{1},x_{2},x_{3})$ gradient ma następującą postać

$\nabla f={\frac {\partial f}{\partial x_{1}}}{\hat {x_{1}}}+{\frac {\partial f}{\partial x_{2}}}{\hat {x_{2}}}+{\frac {\partial f}{\partial x_{3}}}{\hat {x_{3}}}$,

gdzie ${\hat {x_{i}}}$ reprezentuje wektor jednostkowy w kierunku $x_{i}$ dla $1\leq i\leq 3$. Ten rodzaj uogólnionej pochodnej może być rozumiany jako pochodna skalara f po wektorze $\mathbf {x}$ i jego wynik można zebrać w wektor.

$\nabla f={\frac {\partial f}{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}&{\frac {\partial f}{\partial x_{2}}}&{\frac {\partial f}{\partial x_{3}}}\\\end{bmatrix}}.$

More complicated examples include the derivative of a scalar function with respect to a matrix, known as the gradient matrix, which collects the derivative with respect to each matrix element in the corresponding position in the resulting matrix. In that case the scalar must be a function of each of the independent variables in the matrix. As another example, if we have an n-vector of dependent variables, or functions, of m independent variables we might consider the derivative of the dependent vector with respect to the independent vector. The result could be collected in an m×n matrix consisting of all of the possible derivative combinations. There are, of course, a total of nine possibilities using scalars, vectors, and matrices. Notice that as we consider higher numbers of components in each of the independent and dependent variables we can be left with a very large number of possibilities.

The six kinds of derivatives that can be most neatly organized in matrix form are collected in the following table.

Types of Matrix Derivatives

Types	Scalar	Vector	Matrix
Scalar	${\frac {\partial y}{\partial x}}$	${\frac {\partial \mathbf {y} }{\partial x}}$	${\frac {\partial \mathbf {Y} }{\partial x}}$
Vector	${\frac {\partial y}{\partial \mathbf {x} }}$	${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}$
Matrix	${\frac {\partial y}{\partial \mathbf {X} }}$

Here, we have used the term "matrix" in its most general sense, recognizing that vectors and scalars are simply matrices with one column and then one row respectively. Moreover, we have used bold letters to indicate vectors and bold capital letters for matrices. This notation is used throughout.

Notice that we could also talk about the derivative of a vector with respect to a matrix, or any of the other unfilled cells in our table. However, these derivatives are most naturally organized in a tensor of rank higher than 2, so that they do not fit neatly into a matrix. In the following three sections we will define each one of these derivatives and relate them to other branches of mathematics. See the layout conventions section for a more detailed table.

Relation to other derivatives

The matrix derivative is a convenient notation for keeping track of partial derivatives for doing calculations. The Fréchet derivative is the standard way in the setting of functional analysis to take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet differentiable, the two derivatives will agree up to translation of notations. As is the case in general for partial derivatives, some formulae may extend under weaker analytic conditions than the existence of the derivative as approximating linear mapping.

Usages

Matrix calculus is used for deriving optimal stochastic estimators, often involving the use of Lagrange multipliers. This includes the derivation of:

• Kalman filter

• Wiener filter

• Expectation-maximization algorithm for Gaussian mixture

Notation

The vector and matrix derivatives presented in the sections to follow take full advantage of matrix notation, using a single variable to represent a large number of variables. In what follows we will distinguish scalars, vectors and matrices by their typeface. We will let M(n,m) denote the space of real n×m matrices with n rows and m columns. Such matrices will be denoted using bold capital letters: A, X, Y, etc. An element of M(n,1), that is, a column vector, is denoted with a boldface lowercase letter: a, x, y, etc. An element of M(1,1) is a scalar, denoted with lowercase italic typeface: a, t, x, etc. X^T denotes matrix transpose, tr(X) is the trace, and det(X) is the determinant. All functions are assumed to be of differentiability class C¹ unless otherwise noted. Generally letters from first half of the alphabet (a, b, c, …) will be used to denote constants, and from the second half (t, x, y, …) to denote variables.

NOTE: As mentioned above, there are competing notations for laying out systems of partial derivatives in vectors and matrices, and no standard appears to be emerging yet. The next two introductory sections use the numerator layout convention simply for the purposes of convenience, to avoid overly complicating the discussion. The section after them discusses layout conventions in more detail. It is important to realize the following:

1. Despite the use of the terms "numerator layout" and "denominator layout", there are actually more than two possible notational choices involved. The reason is that the choice of numerator vs. denominator (or in some situations, numerator vs. mixed) can be made independently for scalar-by-vector, vector-by-scalar, vector-by-vector, and scalar-by-matrix derivatives, and a number of authors mix and match their layout choices in various ways.

1. The choice of numerator layout in the introductory sections below does not imply that this is the "correct" or "superior" choice. There are advantages and disadvantages to the various layout types. Serious mistakes can result from carelessly combining formulas written in different layouts, and converting from one layout to another requires care to avoid errors. As a result, when working with existing formulas the best policy is probably to identify whichever layout is used and maintain consistency with it, rather than attempting to use the same layout in all situations.

Alternatives

The tensor index notation with its Einstein summation convention is very similar to the matrix calculus, except one writes only a single component at a time. It has the advantage that one can easily manipulate arbitrarily high rank tensors, whereas tensors of rank higher than two are quite unwieldy with matrix notation. All of the work here can be done in this notation without use of the single-variable matrix notation. However, many problems in estimation theory and other areas of applied mathematics would result in too many indices to properly keep track of, pointing in favor of matrix calculus in those areas. Also, Einstein notation can be very useful in proving the identities presented here, as an alternative to typical element notation, which can become cumbersome when the explicit sums are carried around. Note that a matrix can be considered a tensor of rank two.

Derivatives with vectors

Because vectors are matrices with only one column, the simplest matrix derivatives are vector derivatives.

The notations developed here can accommodate the usual operations of vector calculus by identifying the space M(n,1) of n-vectors with the Euclidean space Rⁿ, and the scalar M(1,1) is identified with R. The corresponding concept from vector calculus is indicated at the end of each subsection.

NOTE: The discussion in this section assumes the numerator layout convention for pedagogical purposes. Some authors use different conventions. The section on layout conventions discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions.

Vector-by-scalar

The derivative of a vector

$\mathbf {y} ={\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{m}\\\end{bmatrix}}$,

by a scalar x is written (in numerator layout notation) as

${\frac {\partial \mathbf {y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}\\{\frac {\partial y_{2}}{\partial x}}\\\vdots \\{\frac {\partial y_{m}}{\partial x}}\\\end{bmatrix}}.$

In vector calculus the derivative of a vector y with respect to a scalar x is known as the tangent vector of the vector y, ${\frac {\partial \mathbf {y} }{\partial x}}$. Notice here that y:R $\rightarrow$ R^m.

Example Simple examples of this include the velocity vector in Euclidean space, which is the tangent vector of the position vector (considered as a function of time). Also, the acceleration is the tangent vector of the velocity.

Scalar-by-vector

The derivative of a scalar y by a vector

$\mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\\\end{bmatrix}}$,

is written (in numerator layout notation) as

${\frac {\partial y}{\partial \mathbf {x} }}=\left[{\frac {\partial y}{\partial x_{1}}}\ \ {\frac {\partial y}{\partial x_{2}}}\ \ \cdots \ \ {\frac {\partial y}{\partial x_{n}}}\right].$

In vector calculus the gradient of a scalar field y, in the space Rⁿ whose independent coordinates are the components of x is the derivative of a scalar by a vector. In physics, the electric field is the vector gradient of the electric potential.

The directional derivative of a scalar function f(x) of the space vector x in the direction of the unit vector u is defined using the gradient as follows.

$\nabla _{\mathbf {u}}{f}({\mathbf {x}})=\nabla f({\mathbf {x}})\cdot {\mathbf {u}}$

Using the notation just defined for the derivative of a scalar with respect to a vector we can re-write the directional derivative as

$\nabla _{\mathbf {u} }f={\frac {\partial f}{\partial \mathbf {x} }}\mathbf {u} .$

This type of notation will be nice when proving product rules and chain rules that come out looking similar to what we are familiar with for the scalar derivative.

Vector-by-vector

Each of the previous two cases can be considered as an application of the derivative of a vector with respect to a vector, using a vector of size one appropriately. Similarly we will find that the derivatives involving matrices will reduce to derivatives involving vectors in a corresponding way.

The derivative of a vector function (a vector whose components are functions)

$\mathbf {y} ={\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{m}\\\end{bmatrix}}$,

with respect to an input vector,

$\mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\\\end{bmatrix}}$,

is written (in numerator layout notation) as

${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.$

In vector calculus, the derivative of a vector function y with respect to a vector x whose components represent a space is known as the pushforward or differential, or the Jacobian matrix.

The pushforward along a vector function f with respect to vector v in R^m is given by

$d\,\mathbf {f} (\mathbf {v} )={\frac {\partial \mathbf {f} }{\partial \mathbf {x} }}\mathbf {v} .$

Derivatives with matrices

There are two types of derivatives with matrices that can be organized into a matrix of the same size. These are the derivative of a matrix by a scalar and the derivative of a scalar by a matrix respectively. These can be useful in minimization problems found many areas of applied mathematics and have adopted the names tangent matrix and gradient matrix respectively after their analogs for vectors.

NOTE: The discussion in this section assumes the numerator layout convention for pedagogical purposes. Some authors use different conventions. The section on layout conventions discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions.

Matrix-by-scalar

The derivative of a matrix function Y by a scalar x is known as the tangent matrix and is given (in numerator layout notation) by

${\frac {\partial \mathbf {Y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.$

Scalar-by-matrix

The derivative of a scalar y function of a matrix X of independent variables, with respect to the matrix X, is given (in numerator layout notation) by

${\frac {\partial y}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.$

Notice that the indexing of the gradient with respect to X is transposed as compared with the indexing of X. Important examples of scalar functions of matrices include the trace of a matrix and the determinant.

In analog with vector calculus this derivative is often written as the following.

$\nabla _{\mathbf {X} }y(\mathbf {X} )={\frac {\partial y(\mathbf {X} )}{\partial \mathbf {X} }}$

Also in analog with vector calculus, the directional derivative of a scalar f(X) of a matrix X in the direction of matrix Y is given by

$\nabla _{\mathbf {Y} }f=\operatorname {tr} \left({\frac {\partial f}{\partial \mathbf {X} }}\mathbf {Y} \right).$

It is the gradient matrix, in particular, that finds many uses in minimization problems in estimation theory, particularly in the derivation of the Kalman filter algorithm, which is of great importance in the field.

Other matrix derivatives

The three types of derivatives that have not been considered are those involving vectors-by-matrices, matrices-by-vectors, and matrices-by-matrices. These are not as widely considered and a notation is not widely agreed upon. As for vectors, the other two types of higher matrix derivatives can be seen as applications of the derivative of a matrix by a matrix by using a matrix with one column in the correct place. For this reason, in this subsection we consider only how one can write the derivative of a matrix by another matrix.

The differential or the matrix derivative of a matrix function F(X) that maps from n×m matrices to p×q matrices, F : M(n,m) $\to$ M(p,q), is an element of M(p,q) ? M(m,n), a fourth-rank tensor (the reversal of m and n here indicates the dual space of M(n,m)). In short it is an m×n matrix each of whose entries is a p×q matrix.

${\frac {\partial \mathbf {F} }{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial \mathbf {F} }{\partial X_{1,1}}}&\cdots &{\frac {\partial \mathbf {F} }{\partial X_{n,1}}}\\\vdots &\ddots &\vdots \\{\frac {\partial \mathbf {F} }{\partial X_{1,m}}}&\cdots &{\frac {\partial \mathbf {F} }{\partial X_{n,m}}}\\\end{bmatrix}},$

and note that each ${\frac {\partial \mathbf {F} }{\partial \mathbf {X} _{ij}}}$ is a p×q matrix defined as above. Note also that this matrix has its indexing transposed; m rows and n columns. The pushforward along F of an n×m matrix Y in M(n,m) is then

$d\mathbf {F} (\mathbf {Y} )=\operatorname {tr} \left({\frac {\partial \mathbf {F} }{\partial \mathbf {X} }}\mathbf {Y} \right),$ as formal block matrices.

Note that this definition encompasses all of the preceding definitions as special cases.

According to Jan R. Magnus and Heinz Neudecker, the following notations are both unsuitable, as the determinant of the second resulting matrix would have "no interpretation" and "a useful chain rule does not exist" if these notations are being used:

Given $\phi$, a differentiable function of an $n\times m$ matrix $\mathbf {X} =(x_{i,j})$,

${\frac {\partial \mathbf {\phi } (\mathbf {X} )}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial \mathbf {\phi } }{\partial x_{1,1}}}&\cdots &{\frac {\partial \mathbf {\phi } }{\partial x_{1,q}}}\\\vdots &\ddots &\vdots \\{\frac {\partial \mathbf {\phi } }{\partial x_{n,1}}}&\cdots &{\frac {\partial \mathbf {\phi } }{\partial x_{n,q}}}\\\end{bmatrix}}$

Given $\mathbf {F} =(f_{s,t})$, a differentiable $m\times n$ function of an $n\times m$ matrix $\mathbf {X}$,

${\frac {\partial \mathbf {F} (\mathbf {X} )}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial f_{1,1}}{\partial \mathbf {X} }}&\cdots &{\frac {\partial f_{1,p}}{\partial \mathbf {X} }}\\\vdots &\ddots &\vdots \\{\frac {\partial f_{m,1}}{\partial \mathbf {X} }}&\cdots &{\frac {\partial f_{m,p}}{\partial \mathbf {X} }}\\\end{bmatrix}}$

The Jacobian matrix, according to Magnus and Neudecker, is

$\mathrm {D} \,\mathbf {F} \left(\mathbf {X} \right)={\frac {\partial \,\mathrm {vec} \ \mathbf {F} \left(\mathbf {X} \right)}{\partial \left(\mathrm {vec} \ \mathbf {X} \right)^{\prime }}}.$

Layout conventions

This section discusses the similarities and differences between notational conventions that are used in the various fields that take advantage of matrix calculus. Although there are largely two consistent conventions, some authors find it convenient to mix the two conventions in forms that are discussed below. After this section equations will be listed in both competing forms separately.

The fundamental issue is that the derivative of a vector with respect to a vector, i.e. ${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}$, is often written in two competing ways. If the numerator y is of size m and the denominator x of size n, then the result can be laid out as either an m×n matrix or n×m matrix, i.e. the elements of y laid out in columns and the elements of x laid out in rows, or vice versa. This leads to the following possibilities:

1. Numerator layout, i.e. lay out according to y and x^T (i.e. contrarily to x). This is sometimes known as the Jacobian formulation.

1. Denominator layout, i.e. lay out according to y^T and x (i.e. contrarily to y). This is sometimes known as the Hessian formulation. Some authors term this layout the gradient, in distinction to the Jacobian (numerator layout), which is its transpose. (However, "gradient" more commonly means the derivative ${\frac {\partial y}{\partial \mathbf {x} }},$ regardless of layout.)

1. A third possibility sometimes seen is to insist on writing the derivative as ${\frac {\partial \mathbf {y} }{\partial \mathbf {x} '}},$ (i.e. the derivative is taken with respect to the transpose of x) and follow the numerator layout. This makes it possible to claim that the matrix is laid out according to both numerator and denominator. In practice this produces results the same as the numerator layout.

When handling the gradient ${\frac {\partial y}{\partial \mathbf {x} }}$ and the opposite case ${\frac {\partial \mathbf {y} }{\partial x}},$ we have the same issues. To be consistent, we should do one of the following:

1. If we choose numerator layout for ${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},$ we should lay out the gradient ${\frac {\partial y}{\partial \mathbf {x} }}$ as a row vector, and ${\frac {\partial \mathbf {y} }{\partial x}}$ as a column vector.

1. If we choose denominator layout for ${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},$ we should lay out the gradient ${\frac {\partial y}{\partial \mathbf {x} }}$ as a column vector, and ${\frac {\partial \mathbf {y} }{\partial x}}$ as a row vector.

1. In the third possibility above, we write ${\frac {\partial y}{\partial \mathbf {x} '}}$ and ${\frac {\partial \mathbf {y} }{\partial x}},$ and use numerator layout.

Not all math textbooks and papers are consistent in this respect throughout the entire paper. That is, sometimes different conventions are used in different contexts within the same paper. For example, some choose denominator layout for gradients (laying them out as column vectors), but numerator layout for the vector-by-vector derivative ${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}.$

Similarly, when it comes to scalar-by-matrix derivatives ${\frac {\partial y}{\partial \mathbf {X} }}$ and matrix-by-scalar derivatives ${\frac {\partial \mathbf {Y} }{\partial x}},$ then consistent numerator layout lays out according to Y and X^T, while consistent denominator layout lays out according to Y^T and X. In practice, however, following a denominator layout for ${\frac {\partial \mathbf {Y} }{\partial x}},$ and laying the result out according to Y^T, is rarely seen because it makes for ugly formulas that do not correspond to the scalar formulas. As a result, the following layouts can often be found:

1. Consistent numerator layout, which lays out ${\frac {\partial \mathbf {Y} }{\partial x}}$ according to Y and ${\frac {\partial y}{\partial \mathbf {X} }}$ according to X^T.

1. Mixed layout, which lays out ${\frac {\partial \mathbf {Y} }{\partial x}}$ according to Y and ${\frac {\partial y}{\partial \mathbf {X} }}$ according to X.

1. Use the notation ${\frac {\partial y}{\partial \mathbf {X} ^{\rm {T}}}},$ with results the same as consistent numerator layout.

In the following formulas, we handle the five possible combinations ${\frac {\partial y}{\partial \mathbf {x} }},{\frac {\partial \mathbf {y} }{\partial x}},{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},{\frac {\partial y}{\partial \mathbf {X} }}$ and ${\frac {\partial \mathbf {Y} }{\partial x}}$ separately. We also handle cases of scalar-by-scalar derivatives that involve an intermediate vector or matrix. (This can arise, for example, if a multi-dimensional parametric curve is defined in terms of a scalar variable, and then a derivative of a scalar function of the curve is taken with respect to the scalar that parameterizes the curve.) For each of the various combinations, we give numerator-layout and denominator-layout results, except in the cases above where denominator layout rarely occurs. In cases involving matrices where it makes sense, we give numerator-layout and mixed-layout results. As noted above, cases where vector and matrix denominators are written in transpose notation are equivalent to numerator layout with the denominators written without the transpose.

Keep in mind that various authors use different combinations of numerator and denominator layouts for different types of derivatives, and there is no guarantee that an author will consistently use either numerator or denominator layout for all types. Match up the formulas below with those quoted in the source to determine the layout used for that particular type of derivative, but be careful not to assume that derivatives of other types necessarily follow the same kind of layout.

When taking derivatives with an aggregate (vector or matrix) denominator in order to find a maximum or minimum of the aggregate, it should be kept in mind that using numerator layout will produce results that are transposed with respect to the aggregate. For example, in attempting to find the maximum likelihood estimate of a multivariate normal distribution using matrix calculus, if the domain is a kx1 column vector, then the result using the numerator layout will be in the form of a 1xk row vector. Thus, either the results should be transposed at the end or the denominator layout (or mixed layout) should be used.

Result of differentiating various kinds of aggregates with other kinds of aggregates

	Scalar y		Vector y (size m)		Matrix Y (size m×n)
	Notation	Type	Notation	Type	Notation	Type
Scalar x	${\frac {\partial y}{\partial x}}$	scalar	${\frac {\partial \mathbf {y} }{\partial x}}$	(numerator layout) size-m column vector (denominator layout) size-m row vector	${\frac {\partial \mathbf {Y} }{\partial x}}$	(numerator layout) m×n matrix
Vector x (size n)	${\frac {\partial y}{\partial \mathbf {x} }}$	(numerator layout) size-n row vector (denominator layout) size-n column vector	${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}$	(numerator layout) m×n matrix (denominator layout) n×m matrix	${\frac {\partial \mathbf {Y} }{\partial \mathbf {x} }}$	?
Matrix X (size p×q)	${\frac {\partial y}{\partial \mathbf {X} }}$	(numerator layout) q×p matrix (denominator layout) p×q matrix	${\frac {\partial \mathbf {y} }{\partial \mathbf {X} }}$	?	${\frac {\partial \mathbf {Y} }{\partial \mathbf {X} }}$	?

The results of operations will be transposed when switching between numerator-layout and denominator-layout notation.

Numerator-layout notation

Using numerator-layout notation, we have:

${\frac {\partial y}{\partial \mathbf {x} }}=\left[{\frac {\partial y}{\partial x_{1}}}{\frac {\partial y}{\partial x_{2}}}\cdots {\frac {\partial y}{\partial x_{n}}}\right].$

${\frac {\partial \mathbf {y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}\\{\frac {\partial y_{2}}{\partial x}}\\\vdots \\{\frac {\partial y_{m}}{\partial x}}\\\end{bmatrix}}.$

${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.$

${\frac {\partial y}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.$

The following definitions are only provided in numerator-layout notation:

${\frac {\partial \mathbf {Y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.$

$d\mathbf {X} ={\begin{bmatrix}dx_{11}&dx_{12}&\cdots &dx_{1n}\\dx_{21}&dx_{22}&\cdots &dx_{2n}\\\vdots &\vdots &\ddots &\vdots \\dx_{m1}&dx_{m2}&\cdots &dx_{mn}\\\end{bmatrix}}.$

Denominator-layout notation

Using denominator-layout notation, we have:

${\frac {\partial y}{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}\\{\frac {\partial y}{\partial x_{2}}}\\\vdots \\{\frac {\partial y}{\partial x_{n}}}\\\end{bmatrix}}.$

${\frac {\partial \mathbf {y} }{\partial x}}=\left[{\frac {\partial y_{1}}{\partial x}}{\frac {\partial y_{2}}{\partial x}}\cdots {\frac {\partial y_{m}}{\partial x}}\right].$

${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{1}}}\\{\frac {\partial y_{1}}{\partial x_{2}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{1}}{\partial x_{n}}}&{\frac {\partial y_{2}}{\partial x_{n}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.$

${\frac {\partial y}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{12}}}&\cdots &{\frac {\partial y}{\partial x_{1q}}}\\{\frac {\partial y}{\partial x_{21}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{2q}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{p1}}}&{\frac {\partial y}{\partial x_{p2}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.$

Identities

As noted above, in general, the results of operations will be transposed when switching between numerator-layout and denominator-layout notation.

To help make sense of all the identities below, keep in mind the most important rules: the chain rule, product rule and sum rule. The sum rule applies universally, and the product rule applies in most of the cases below, provided that the order of matrix products is maintained, since matrix products are not commutative. The chain rule applies in some of the cases, but unfortunately does not apply in matrix-by-scalar derivatives or scalar-by-matrix derivatives (in the latter case, mostly involving the trace operator applied to matrices). In the latter case, the product rule can't quite be applied directly, either, but the equivalent can be done with a bit more work using the differential identities.

Vector-by-vector identities

This is presented first because all of the operations that apply to vector-by-vector differentiation apply directly to vector-by-scalar or scalar-by-vector differentiation simply by reducing the appropriate vector in the numerator or denominator to a scalar.

Identities: vector-by-vector ${\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}$

Condition	Expression	Numerator layout, i.e. by y and xT	Denominator layout, i.e. by yT and x
a is not a function of x	${\frac {\partial \mathbf {a} }{\partial \mathbf {x} }}=$	$\mathbf {0}$
	${\frac {\partial \mathbf {x} }{\partial \mathbf {x} }}=$	$\mathbf {I}$
A is not a function of x	${\frac {\partial \mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=$	$\mathbf {A}$	$\mathbf {A} ^{\rm {T}}$
A is not a function of x	${\frac {\partial \mathbf {x} ^{\rm {T}}\mathbf {A} }{\partial \mathbf {x} }}=$	$\mathbf {A} ^{\rm {T}}$	$\mathbf {A}$
a is not a function of x, u = u(x)	${\frac {\partial a\mathbf {u} }{\partial \,\mathbf {x} }}=$	$a{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}$
a = a(x), u = u(x)	${\frac {\partial a\mathbf {u} }{\partial \mathbf {x} }}=$	$a{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+\mathbf {u} {\frac {\partial a}{\partial \mathbf {x} }}$	$a{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial a}{\partial \mathbf {x} }}\mathbf {u} ^{\rm {T}}$
A is not a function of x, u = u(x)	${\frac {\partial \mathbf {A} \mathbf {u} }{\partial \mathbf {x} }}=$	$\mathbf {A} {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}$	${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} ^{\rm {T}}$
u = u(x), v = v(x)	${\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial \mathbf {x} }}=$	${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}$
u = u(x)	${\frac {\partial \mathbf {g(u)} }{\partial \mathbf {x} }}=$	${\frac {\partial \mathbf {g(u)} }{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}$	${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g(u)} }{\partial \mathbf {u} }}$
u = u(x)	${\frac {\partial \mathbf {f(g(u))} }{\partial \mathbf {x} }}=$	${\frac {\partial \mathbf {f(g)} }{\partial \mathbf {g} }}{\frac {\partial \mathbf {g(u)} }{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}$	${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g(u)} }{\partial \mathbf {u} }}{\frac {\partial \mathbf {f(g)} }{\partial \mathbf {g} }}$

Scalar-by-vector identities

The fundamental identities are placed above the thick black line.

Identities: scalar-by-vector ${\frac {\partial y}{\partial \mathbf {x} }}=\nabla _{\mathbf {x} }y$

Condition	Expression	Numerator layout, i.e. by xT; result is row vector	Denominator layout, i.e. by x; result is column vector
a is not a function of x	${\frac {\partial a}{\partial \mathbf {x} }}=$	$\mathbf {0} ^{\rm {T}}$	$\mathbf {0}$
a is not a function of x, u = u(x)	${\frac {\partial au}{\partial \mathbf {x} }}=$	$a{\frac {\partial u}{\partial \mathbf {x} }}$
u = u(x), v = v(x)	${\frac {\partial (u+v)}{\partial \mathbf {x} }}=$	${\frac {\partial u}{\partial \mathbf {x} }}+{\frac {\partial v}{\partial \mathbf {x} }}$
u = u(x), v = v(x)	${\frac {\partial uv}{\partial \mathbf {x} }}=$	$u{\frac {\partial v}{\partial \mathbf {x} }}+v{\frac {\partial u}{\partial \mathbf {x} }}$
u = u(x)	${\frac {\partial g(u)}{\partial \mathbf {x} }}=$	${\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}$
u = u(x)	${\frac {\partial f(g(u))}{\partial \mathbf {x} }}=$	${\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}$
u = u(x), v = v(x)	${\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\rm {T}}\mathbf {v} }{\partial \mathbf {x} }}=$	$\mathbf {u} ^{\rm {T}}{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\rm {T}}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}$ assumes numerator layout of ${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}$	${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {u}$ assumes denominator layout of ${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}$
u = u(x), v = v(x), A is not a function of x	${\frac {\partial (\mathbf {u} \cdot \mathbf {A} \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\rm {T}}\mathbf {A} \mathbf {v} }{\partial \mathbf {x} }}=$	$\mathbf {u} ^{\rm {T}}\mathbf {A} {\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\rm {T}}\mathbf {A} ^{\rm {T}}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}$ assumes numerator layout of ${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}$	${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} \mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {A} ^{\rm {T}}\mathbf {u}$ assumes denominator layout of ${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}$
a is not a function of x	${\frac {\partial (\mathbf {a} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial (\mathbf {x} \cdot \mathbf {a} )}{\partial \mathbf {x} }}=$ ${\frac {\partial \mathbf {a} ^{\rm {T}}\mathbf {x} }{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\rm {T}}\mathbf {a} }{\partial \mathbf {x} }}=$	$\mathbf {a} ^{\rm {T}}$	$\mathbf {a}$
A is not a function of x b is not a function of x	${\frac {\partial \mathbf {b} ^{\rm {T}}\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=$	$\mathbf {b} ^{\rm {T}}\mathbf {A}$	$\mathbf {A} ^{\rm {T}}\mathbf {b}$
A is not a function of x	${\frac {\partial \mathbf {x} ^{\rm {T}}\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=$	$\mathbf {x} ^{\rm {T}}(\mathbf {A} +\mathbf {A} ^{\rm {T}})$	$(\mathbf {A} +\mathbf {A} ^{\rm {T}})\mathbf {x}$
A is not a function of x A is symmetric	${\frac {\partial \mathbf {x} ^{\rm {T}}\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=$	$2\mathbf {x} ^{\rm {T}}\mathbf {A}$	$2\mathbf {A} \mathbf {x}$
A is not a function of x	${\frac {\partial ^{2}\mathbf {x} ^{\rm {T}}\mathbf {A} \mathbf {x} }{\partial \mathbf {x} ^{2}}}=$	$\mathbf {A} +\mathbf {A} ^{\rm {T}}$
A is not a function of x A is symmetric	${\frac {\partial ^{2}\mathbf {x} ^{\rm {T}}\mathbf {A} \mathbf {x} }{\partial \mathbf {x} ^{2}}}=$	$2\mathbf {A}$
	${\frac {\partial (\mathbf {x} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\rm {T}}\mathbf {x} }{\partial \mathbf {x} }}=$	$2\mathbf {x} ^{\rm {T}}$	$2\mathbf {x}$
a is not a function of x, u = u(x)	${\frac {\partial (\mathbf {a} \cdot \mathbf {u} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {a} ^{\rm {T}}\mathbf {u} }{\partial \mathbf {x} }}=$	$\mathbf {a} ^{\rm {T}}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}$ assumes numerator layout of ${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}$	${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {a}$ assumes denominator layout of ${\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}$
a, b are not functions of x	${\frac {\partial \;{\textbf {a}}^{\rm {T}}{\textbf {x}}{\textbf {x}}^{\rm {T}}{\textbf {b}}}{\partial \;{\textbf {x}}}}=$	${\textbf {x}}^{\rm {T}}({\textbf {a}}{\textbf {b}}^{\rm {T}}+{\textbf {b}}{\textbf {a}}^{\rm {T}})$	$({\textbf {a}}{\textbf {b}}^{\rm {T}}+{\textbf {b}}{\textbf {a}}^{\rm {T}}){\textbf {x}}$
A, b, C, D, e are not functions of x	${\frac {\partial \;({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\rm {T}}{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})}{\partial \;{\textbf {x}}}}=$	$({\textbf {D}}{\textbf {x}}+{\textbf {e}})^{\rm {T}}{\textbf {C}}^{\rm {T}}{\textbf {A}}+({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\rm {T}}{\textbf {C}}{\textbf {D}}$	${\textbf {D}}^{\rm {T}}{\textbf {C}}^{\rm {T}}({\textbf {A}}{\textbf {x}}+{\textbf {b}})+{\textbf {A}}^{\rm {T}}{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})$
a is not a function of x	${\frac {\partial \;\|\mathbf {x} -\mathbf {a} \|}{\partial \;\mathbf {x} }}=$	${\frac {(\mathbf {x} -\mathbf {a} )^{\rm {T}}}{\|\mathbf {x} -\mathbf {a} \|}}$	${\frac {\mathbf {x} -\mathbf {a} }{\|\mathbf {x} -\mathbf {a} \|}}$

Vector-by-scalar identities

Identities: vector-by-scalar ${\frac {\partial \mathbf {y} }{\partial x}}$

Condition	Expression	Numerator layout, i.e. by y, result is column vector	Denominator layout, i.e. by yT, result is row vector
a is not a function of x	${\frac {\partial \mathbf {a} }{\partial x}}=$	$\mathbf {0}$
a is not a function of x, u = u(x)	${\frac {\partial a\mathbf {u} }{\partial x}}=$	$a{\frac {\partial \mathbf {u} }{\partial x}}$
A is not a function of x, u = u(x)	${\frac {\partial \mathbf {A} \mathbf {u} }{\partial x}}=$	$\mathbf {A} {\frac {\partial \mathbf {u} }{\partial x}}$	${\frac {\partial \mathbf {u} }{\partial x}}\mathbf {A} ^{\rm {T}}$
u = u(x)	${\frac {\partial \mathbf {u} ^{\rm {T}}}{\partial x}}=$	$\left({\frac {\partial \mathbf {u} }{\partial x}}\right)^{\rm {T}}$
u = u(x), v = v(x)	${\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial x}}=$	${\frac {\partial \mathbf {u} }{\partial x}}+{\frac {\partial \mathbf {v} }{\partial x}}$
u = u(x), v = v(x)	${\frac {\partial (\mathbf {u} \times \mathbf {v} )}{\partial x}}=$	$\mathbf {u} \times {\frac {\partial \mathbf {v} }{\partial x}}+{\frac {\partial \mathbf {u} }{\partial x}}\times \mathbf {v}$
u = u(x)	${\frac {\partial \mathbf {g(u)} }{\partial x}}=$	${\frac {\partial \mathbf {g(u)} }{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}$	${\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g(u)} }{\partial \mathbf {u} }}$
		Assumes consistent matrix layout; see below.
u = u(x)	${\frac {\partial \mathbf {f(g(u))} }{\partial x}}=$	${\frac {\partial \mathbf {f(g)} }{\partial \mathbf {g} }}{\frac {\partial \mathbf {g(u)} }{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}$	${\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g(u)} }{\partial \mathbf {u} }}{\frac {\partial \mathbf {f(g)} }{\partial \mathbf {g} }}$
		Assumes consistent matrix layout; see below.

NOTE: The formulas involving the vector-by-vector derivatives ${\frac {\partial \mathbf {g(u)} }{\partial \mathbf {u} }}$ and ${\frac {\partial \mathbf {f(g)} }{\partial \mathbf {g} }}$ (whose outputs are matrices) assume the matrices are laid out consistent with the vector layout, i.e. numerator-layout matrix when numerator-layout vector and vice versa; otherwise, transpose the vector-by-vector derivatives.

Scalar-by-matrix identities

Note that exact equivalents of the scalar product rule and chain rule do not exist when applied to matrix-valued functions of matrices. However, the product rule of this sort does apply to the differential form (see below), and this is the way to derive many of the identities below involving the trace function, combined with the fact that the trace function allows transposing and cyclic permutation, i.e.:

${\rm {tr}}(\mathbf {A} )={\rm {tr}}(\mathbf {A^{\rm {T}}} )$

${\rm {tr}}(\mathbf {ABCD} )={\rm {tr}}(\mathbf {BCDA} )={\rm {tr}}(\mathbf {CDAB} )={\rm {tr}}(\mathbf {DABC} )$

For example, to compute ${\frac {\partial {\rm {tr}}(\mathbf {AXBX^{\rm {T}}C} )}{\partial \mathbf {X} }}:$

${\begin{aligned}d\,{\rm {tr}}(\mathbf {AXBX^{\rm {T}}C} )&=d\,{\rm {tr}}(\mathbf {CAXBX^{\rm {T}}} )={\rm {tr}}(d(\mathbf {CAXBX^{\rm {T}}} ))\\&={\rm {tr}}(\mathbf {CAX} d(\mathbf {BX^{\rm {T}}} )+d(\mathbf {CAX} )\mathbf {BX^{\rm {T}}} )\\&={\rm {tr}}(\mathbf {CAX} d(\mathbf {BX^{\rm {T}}} ))+{\rm {tr}}(d(\mathbf {CAX} )\mathbf {BX^{\rm {T}}} )\\&={\rm {tr}}(\mathbf {CAXB} d(\mathbf {X^{\rm {T}}} ))+{\rm {tr}}(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\rm {T}}} )\\&={\rm {tr}}(\mathbf {CAXB} (d\mathbf {X} )^{\rm {T}})+{\rm {tr}}(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\rm {T}}} )\\&={\rm {tr}}\left((\mathbf {CAXB} (d\mathbf {X} )^{\rm {T}})^{\rm {T}}\right)+{\rm {tr}}(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\rm {T}}} )\\&={\rm {tr}}((d\mathbf {X} )\mathbf {B^{\rm {T}}X^{\rm {T}}A^{\rm {T}}C^{\rm {T}}} )+{\rm {tr}}(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\rm {T}}} )\\&={\rm {tr}}(\mathbf {B^{\rm {T}}X^{\rm {T}}A^{\rm {T}}C^{\rm {T}}} (d\mathbf {X} ))+{\rm {tr}}(\mathbf {BX^{\rm {T}}} \mathbf {CA} (d\mathbf {X} ))\\&={\rm {tr}}\left((\mathbf {B^{\rm {T}}X^{\rm {T}}A^{\rm {T}}C^{\rm {T}}} +\mathbf {BX^{\rm {T}}} \mathbf {CA} )d\mathbf {X} \right)\end{aligned}}$

Therefore,

${\frac {\partial {\rm {tr}}(\mathbf {AXBX^{\rm {T}}C} )}{\partial \mathbf {X} }}=\mathbf {B^{\rm {T}}X^{\rm {T}}A^{\rm {T}}C^{\rm {T}}} +\mathbf {BX^{\rm {T}}} \mathbf {CA} .$

(For the last step, see the `Conversion from differential to derivative form' section.)

Identities: scalar-by-matrix ${\frac {\partial y}{\partial \mathbf {X} }}$

Condition	Expression	Numerator layout, i.e. by XT	Denominator layout, i.e. by X
a is not a function of X	${\frac {\partial a}{\partial \mathbf {X} }}=$	$\mathbf {0} ^{\rm {T}}$	$\mathbf {0}$
a is not a function of X, u = u(X)	${\frac {\partial au}{\partial \mathbf {X} }}=$	$a{\frac {\partial u}{\partial \mathbf {X} }}$
u = u(X), v = v(X)	${\frac {\partial (u+v)}{\partial \mathbf {X} }}=$	${\frac {\partial u}{\partial \mathbf {X} }}+{\frac {\partial v}{\partial \mathbf {X} }}$
u = u(X), v = v(X)	${\frac {\partial uv}{\partial \mathbf {X} }}=$	$u{\frac {\partial v}{\partial \mathbf {X} }}+v{\frac {\partial u}{\partial \mathbf {X} }}$
u = u(X)	${\frac {\partial g(u)}{\partial \mathbf {X} }}=$	${\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}$
u = u(X)	${\frac {\partial f(g(u))}{\partial \mathbf {X} }}=$	${\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}$
U = U(X)	${\frac {\partial g(\mathbf {U} )}{\partial X_{ij}}}=$	${\rm {tr}}\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)$	${\rm {tr}}\left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\rm {T}}{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)$
		Both forms assume numerator layout for ${\frac {\partial \mathbf {U} }{\partial X_{ij}}},$ i.e. mixed layout if denominator layout for X is being used.
	${\frac {\partial {\rm {tr}}(\mathbf {X} )}{\partial \mathbf {X} }}=$	$\mathbf {I}$
U = U(X), V = V(X)	${\frac {\partial {\rm {tr}}(\mathbf {U} +\mathbf {V} )}{\partial \mathbf {X} }}=$	${\frac {\partial {\rm {tr}}(\mathbf {U} )}{\partial \mathbf {X} }}+{\frac {\partial {\rm {tr}}(\mathbf {V} )}{\partial \mathbf {X} }}$
a is not a function of X, U = U(X)	${\frac {\partial {\rm {tr}}(a\mathbf {U} )}{\partial \mathbf {X} }}=$	$a{\frac {\partial {\rm {tr}}(\mathbf {U} )}{\partial \mathbf {X} }}$
g(X) is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. eX, sin(X), cos(X), ln(X), etc. using a Taylor series); g(x) is the equivalent scalar function, g′(x) is its derivative, and g′(X) is the corresponding matrix function	${\frac {\partial {\rm {tr}}(\mathbf {g(X)} )}{\partial \mathbf {X} }}=$	$\mathbf {g} '(\mathbf {X} )$	$(\mathbf {g} '(\mathbf {X} ))^{\rm {T}}$
A is not a function of X	${\frac {\partial {\rm {tr}}(\mathbf {AX} )}{\partial \mathbf {X} }}={\frac {\partial {\rm {tr}}(\mathbf {XA} )}{\partial \mathbf {X} }}=$	$\mathbf {A}$	$\mathbf {A} ^{\rm {T}}$
A is not a function of X	${\frac {\partial {\rm {tr}}(\mathbf {AX^{\rm {T}}} )}{\partial \mathbf {X} }}={\frac {\partial {\rm {tr}}(\mathbf {X^{\rm {T}}A} )}{\partial \mathbf {X} }}=$	$\mathbf {A} ^{\rm {T}}$	$\mathbf {A}$
A is not a function of X	${\frac {\partial {\rm {tr}}(\mathbf {X^{\rm {T}}AX} )}{\partial \mathbf {X} }}=$	$\mathbf {X} ^{\rm {T}}(\mathbf {A} +\mathbf {A} ^{\rm {T}})$	$(\mathbf {A} +\mathbf {A} ^{\rm {T}})\mathbf {X}$
A is not a function of X	${\frac {\partial {\rm {tr}}(\mathbf {X^{-1}A} )}{\partial \mathbf {X} }}=$	$-(\mathbf {X} ^{-1})^{\rm {T}}\mathbf {A} (\mathbf {X} ^{-1})^{\rm {T}}$	$-\mathbf {X} ^{-1}\mathbf {A} ^{\rm {T}}\mathbf {X} ^{-1}$
A, B are not functions of X	${\frac {\partial {\rm {tr}}(\mathbf {AXB} )}{\partial \mathbf {X} }}={\frac {\partial {\rm {tr}}(\mathbf {BAX} )}{\partial \mathbf {X} }}=$	$\mathbf {BA}$	$\mathbf {A^{\rm {T}}B^{\rm {T}}}$
A, B, C are not functions of X	${\frac {\partial {\rm {tr}}(\mathbf {AXBX^{\rm {T}}C} )}{\partial \mathbf {X} }}=$	$\mathbf {BX^{\rm {T}}CA} +\mathbf {B^{\rm {T}}X^{\rm {T}}A^{\rm {T}}C^{\rm {T}}}$	$\mathbf {A^{\rm {T}}C^{\rm {T}}XB^{\rm {T}}} +\mathbf {CAXB}$
n is a positive integer	${\frac {\partial {\rm {tr}}(\mathbf {X} ^{n})}{\partial \mathbf {X} }}=$	$n\mathbf {X} ^{n-1}$	$n(\mathbf {X} ^{n-1})^{\rm {T}}$
A is not a function of X, n is a positive integer	${\frac {\partial {\rm {tr}}(\mathbf {A} \mathbf {X} ^{n})}{\partial \mathbf {X} }}=$	$\sum _{i=0}^{n-1}\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1}$	$\sum _{i=0}^{n-1}(\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1})^{\rm {T}}$
	${\frac {\partial {\rm {tr}}(e^{\mathbf {X} })}{\partial \mathbf {X} }}=$	$e^{\mathbf {X} }$	$(e^{\mathbf {X} })^{\rm {T}}$
	${\frac {\partial {\rm {tr}}(\sin(\mathbf {X} ))}{\partial \mathbf {X} }}=$	$\cos(\mathbf {X} )$	$(\cos(\mathbf {X} ))^{\rm {T}}$
	${\frac {\partial |\mathbf {X} |}{\partial \mathbf {X} }}=$	$\operatorname {cofactor} (X)^{\rm {T}}=|\mathbf {X} |\mathbf {X} ^{-1}$	$\operatorname {cofactor} (X)=|\mathbf {X} |(\mathbf {X} ^{-1})^{\rm {T}}$
a is not a function of X	${\frac {\partial \ln |a\mathbf {X} |}{\partial \mathbf {X} }}=$	$\mathbf {X} ^{-1}$	$(\mathbf {X} ^{-1})^{\rm {T}}$
A, B are not functions of X	${\frac {\partial |\mathbf {AXB} |}{\partial \mathbf {X} }}=$	$|\mathbf {AXB} |\mathbf {X} ^{-1}$	$|\mathbf {AXB} |(\mathbf {X} ^{-1})^{\rm {T}}$
n is a positive integer	${\frac {\partial |\mathbf {X} ^{n}|}{\partial \mathbf {X} }}=$	$n|\mathbf {X} ^{n}|\mathbf {X} ^{-1}$	$n|\mathbf {X} ^{n}|(\mathbf {X} ^{-1})^{\rm {T}}$
(see pseudo-inverse)	${\frac {\partial \ln |\mathbf {X} ^{\rm {T}}\mathbf {X} |}{\partial \mathbf {X} }}=$	$2\mathbf {X} ^{+}$	$2(\mathbf {X} ^{+})^{\rm {T}}$
(see pseudo-inverse)	${\frac {\partial \ln |\mathbf {X} ^{\rm {T}}\mathbf {X} |}{\partial \mathbf {X} ^{+}}}=$	$-2\mathbf {X}$	$-2\mathbf {X} ^{\rm {T}}$
A is not a function of X, X is square and invertible	${\frac {\partial |\mathbf {X^{\rm {T}}} \mathbf {A} \mathbf {X} |}{\partial \mathbf {X} }}=$	$2|\mathbf {X^{\rm {T}}} \mathbf {A} \mathbf {X} |\mathbf {X} ^{-1}$	$2|\mathbf {X^{\rm {T}}} \mathbf {A} \mathbf {X} |(\mathbf {X} ^{-1})^{\rm {T}}$
A is not a function of X, X is non-square, A is symmetric	${\frac {\partial |\mathbf {X^{\rm {T}}} \mathbf {A} \mathbf {X} |}{\partial \mathbf {X} }}=$	$2|\mathbf {X^{\rm {T}}} \mathbf {A} \mathbf {X} |(\mathbf {X^{\rm {T}}A^{\rm {T}}X} )^{-1}\mathbf {X^{\rm {T}}A^{\rm {T}}}$	$2|\mathbf {X^{\rm {T}}} \mathbf {A} \mathbf {X} |\mathbf {AX} (\mathbf {X^{\rm {T}}AX} )^{-1}$
A is not a function of X, X is non-square, A is non-symmetric	${\frac {\partial |\mathbf {X^{\rm {T}}} \mathbf {A} \mathbf {X} |}{\partial \mathbf {X} }}=$	$|\mathbf {X^{\rm {T}}} \mathbf {A} \mathbf {X} |((\mathbf {X^{\rm {T}}AX} )^{-1}\mathbf {X^{\rm {T}}A}$ $+(\mathbf {X^{\rm {T}}A^{\rm {T}}X} )^{-1}\mathbf {X^{\rm {T}}A^{\rm {T}}} )$	$|\mathbf {X^{\rm {T}}} \mathbf {A} \mathbf {X} |(\mathbf {AX} (\mathbf {X^{\rm {T}}AX} )^{-1}$ $+\mathbf {A^{\rm {T}}X} (\mathbf {X^{\rm {T}}A^{\rm {T}}X} )^{-1})$

Matrix-by-scalar identities

Identities: matrix-by-scalar ${\frac {\partial \mathbf {Y} }{\partial x}}$

Condition	Expression	Numerator layout, i.e. by Y
U = U(x)	${\frac {\partial a\mathbf {U} }{\partial x}}=$	$a{\frac {\partial \mathbf {U} }{\partial x}}$
A, B are not functions of x, U = U(x)	${\frac {\partial \mathbf {AUB} }{\partial x}}=$	$\mathbf {A} {\frac {\partial \mathbf {U} }{\partial x}}\mathbf {B}$
U = U(x), V = V(x)	${\frac {\partial (\mathbf {U} +\mathbf {V} )}{\partial x}}=$	${\frac {\partial \mathbf {U} }{\partial x}}+{\frac {\partial \mathbf {V} }{\partial x}}$
U = U(x), V = V(x)	${\frac {\partial (\mathbf {U} \mathbf {V} )}{\partial x}}=$	$\mathbf {U} {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {V}$
U = U(x), V = V(x)	${\frac {\partial (\mathbf {U} \otimes \mathbf {V} )}{\partial x}}=$	$\mathbf {U} \otimes {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\otimes \mathbf {V}$
U = U(x), V = V(x)	${\frac {\partial (\mathbf {U} \circ \mathbf {V} )}{\partial x}}=$	$\mathbf {U} \circ {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\circ \mathbf {V}$
U = U(x)	${\frac {\partial \mathbf {U} ^{-1}}{\partial x}}=$	$-\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}$
U = U(x,y)	${\frac {\partial ^{2}\mathbf {U} ^{-1}}{\partial x\partial y}}=$	$\mathbf {U} ^{-1}\left({\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial y}}-{\frac {\partial ^{2}\mathbf {U} }{\partial x\partial y}}+{\frac {\partial \mathbf {U} }{\partial y}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)\mathbf {U} ^{-1}$
A is not a function of x, g(X) is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. eX, sin(X), cos(X), ln(X), etc.); g(x) is the equivalent scalar function, g′(x) is its derivative, and g′(X) is the corresponding matrix function	${\frac {\partial \,\mathbf {g} (x\mathbf {A} )}{\partial x}}=$	$\mathbf {A} \mathbf {g} '(x\mathbf {A} )=\mathbf {g} '(x\mathbf {A} )\mathbf {A}$
A is not a function of x	${\frac {\partial e^{x\mathbf {A} }}{\partial x}}=$	$\mathbf {A} e^{x\mathbf {A} }=e^{x\mathbf {A} }\mathbf {A}$

Further see Derivative of the exponential map.

Scalar-by-scalar identities

With vectors involved

Identities: scalar-by-scalar, with vectors involved

Condition	Expression	Any layout (assumes dot product ignores row vs. column layout)
u = u(x)	${\frac {\partial g(\mathbf {u} )}{\partial x}}=$	${\frac {\partial g(\mathbf {u} )}{\partial \mathbf {u} }}\cdot {\frac {\partial \mathbf {u} }{\partial x}}$
u = u(x), v = v(x)	${\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial x}}=$	$\mathbf {u} \cdot {\frac {\partial \mathbf {v} }{\partial x}}+{\frac {\partial \mathbf {u} }{\partial x}}\cdot \mathbf {v}$

With matrices involved

Identities: scalar-by-scalar, with matrices involved

Condition	Expression	Consistent numerator layout, i.e. by Y and XT	Mixed layout, i.e. by Y and X
U = U(x)	${\frac {\partial |\mathbf {U} |}{\partial x}}=$	$|\mathbf {U} |{\rm {tr}}\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)$
U = U(x)	${\frac {\partial \ln |\mathbf {U} |}{\partial x}}=$	${\rm {tr}}\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)$
U = U(x)	${\frac {\partial ^{2}|\mathbf {U} |}{\partial x^{2}}}=$	$|\mathbf {U} |\left[{\rm {tr}}\left(\mathbf {U} ^{-1}{\frac {\partial ^{2}\mathbf {U} }{\partial x^{2}}}\right)+\left({\rm {tr}}\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)\right)^{2}-{\rm {tr}}\left(\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)\right)\right]$
U = U(x)	${\frac {\partial g(\mathbf {U} )}{\partial x}}=$	${\rm {tr}}\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial x}}\right)$	${\rm {tr}}\left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\rm {T}}{\frac {\partial \mathbf {U} }{\partial x}}\right)$
A is not a function of x, g(X) is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. eX, sin(X), cos(X), ln(X), etc.); g(x) is the equivalent scalar function, g′(x) is its derivative, and g′(X) is the corresponding matrix function.	${\frac {\partial \,{\rm {tr}}(\mathbf {g} (x\mathbf {A} ))}{\partial x}}=$	${\rm {tr}}(\mathbf {A} \mathbf {g} '(x\mathbf {A} ))$
A is not a function of x	${\frac {\partial \,{\rm {tr}}(e^{x\mathbf {A} })}{\partial x}}=$	${\rm {tr}}(\mathbf {A} e^{x\mathbf {A} })$

Identities in differential form

It is often easier to work in differential form and then convert back to normal derivatives. This only works well using the numerator layout.

Differential identities: scalar involving matrix

Condition	Expression	Result (numerator layout)
	$d({\rm {tr}}(\mathbf {X} ))=$	${\rm {tr}}(d\mathbf {X} )$
	$d(|\mathbf {X} |)=$	$|\mathbf {X} |{\rm {tr}}(\mathbf {X} ^{-1}d\mathbf {X} )$
	$d(\ln |\mathbf {X} |)=$	${\rm {tr}}(\mathbf {X} ^{-1}d\mathbf {X} )$

Differential identities: matrix

Condition	Expression	Result (numerator layout)
A is not a function of X	$d(\mathbf {A} )=$	$0$
a is not a function of X	$d(a\mathbf {X} )=$	$a\,d\mathbf {X}$
	$d(\mathbf {X} +\mathbf {Y} )=$	$d\mathbf {X} +d\mathbf {Y}$
	$d(\mathbf {X} \mathbf {Y} )=$	$(d\mathbf {X} )\mathbf {Y} +\mathbf {X} (d\mathbf {Y} )$
(Kronecker product)	$d(\mathbf {X} \otimes \mathbf {Y} )=$	$(d\mathbf {X} )\otimes \mathbf {Y} +\mathbf {X} \otimes (d\mathbf {Y} )$
(Hadamard product)	$d(\mathbf {X} \circ \mathbf {Y} )=$	$(d\mathbf {X} )\circ \mathbf {Y} +\mathbf {X} \circ (d\mathbf {Y} )$
	$d(\mathbf {X} ^{\rm {T}})=$	$(d\mathbf {X} )^{\rm {T}}$
(conjugate transpose)	$d(\mathbf {X} ^{\rm {H}})=$	$(d\mathbf {X} )^{\rm {H}}$

To convert to normal derivative form, first convert it to one of the following canonical forms, and then use these identities:

Conversion from differential to derivative form

Canonical differential form	Equivalent derivative form
$dy=a\,dx$	${\frac {dy}{dx}}=a$
$dy=\mathbf {a} \,d\mathbf {x}$	${\frac {dy}{d\mathbf {x} }}=\mathbf {a}$
$dy={\rm {tr}}(\mathbf {A} \,d\mathbf {X} )$	${\frac {dy}{d\mathbf {X} }}=\mathbf {A}$
$d\mathbf {y} =\mathbf {a} \,dx$	${\frac {d\mathbf {y} }{dx}}=\mathbf {a}$
$d\mathbf {y} =\mathbf {A} \,d\mathbf {x}$	${\frac {d\mathbf {y} }{d\mathbf {x} }}=\mathbf {A}$
$d\mathbf {Y} =\mathbf {A} \,dx$	${\frac {d\mathbf {Y} }{dx}}=\mathbf {A}$

External links

• Linear Algebra: Determinants, Inverses, Rank appendix D from Introduction to Finite Element Methods book on University of Colorado at Boulder. Uses the Hessian (transpose to Jacobian) definition of vector and matrix derivatives.

• Matrix Reference Manual, Mike Brookes, Imperial College London.

• The Matrix Cookbook (2006), with a derivatives chapter. Uses the Hessian definition.

• The Matrix Cookbook (2012), an updated version of the Matrix Cookbook.

• Linear Algebra and its Applications (author information page; see Chapter 9 of book), Peter Lax, Courant Institute.

• Matrix Differentiation (and some other stuff), Randal J. Barnes, Department of Civil Engineering, University of Minnesota.

• Notes on Matrix Calculus, Paul L. Fackler, North Carolina State University.

• Matrix Differential Calculus (slide presentation), Zhang Le, University of Edinburgh.

• Introduction to Vector and Matrix Differentiation (notes on matrix differentiation, in the context of Econometrics), Heino Bohn Nielsen.

• A note on differentiating matrices (notes on matrix differentiation), Pawel Koval, from Munich Personal RePEc Archive.

• Vector/Matrix Calculus More notes on matrix differentiation.

• Matrix Identities (notes on matrix differentiation), Sam Roweis.

Alfabet łaciński

Szablon:Latin alphabet/main= watykańskie monety euro=

Abbildungen der vatikanischen Euromünzen • Nationale Seite (2014)

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1,00 €	2,00 €	Rand der 2-€-Münze
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Papst Franziskus		Sechsmal abwechselnd die Zahl 2 und ★ (ein Stern), abwechselnd aufrecht und um 180° gedreht.

42.zip

Znaną bombą jest plik 42.zip, który jest spakowany pięciokrotnie rekurencyjnie. Plik ma zaledwie 42 kB wielkości. Przy rozpakowywaniu rozmiar danych rośnie sto miliardów razy do 4.5PB.

struktura                           całkowity    zawiera      nieskompresowana wielkość danych
                                    rozmiar1)    w każdym przypadku      (B)
42.zip2)                             1.048.576   16 katalogów  4.503.599.626.321.920 (4,5 PB)
  → lib0.zip     … libf.zip3)        1.048.576   16 katalogów 4.503.599.626.321.920 (4,5 PB)
   → book0.zip    … bookf.zip          65.536   16 Ordner      281.474.976.645.120 (281 TB)
    → chapter0.zip … chapterf.zip       4.096   16 Ordner       17.592.186.040.320 ( 17 TB)
     → doc0.zip     … docf.zip            256   16 Ordner        1.099.511.627.520 (  1 TB)
      → page0.zip    … pagef.zip           16    1 Datei            68.719.476.720 ( 68 GB)
       → 0.dll4)                             1                        4.294.967.295 (4,3 GB)

Anmerkungen:

• ¹⁾ Ausschließlich die unterste Hierarchiestufe – also die Dateien page0.zip … pagef.zip – enthält Dateien; die darüber Liegenden dienen dienen nur als Multiplikatoren für die Anzahl der Dateien in der untersten Hierarchiestufe. Die Angaben sind nicht kumuliert und dienen vornehmlich der Anschauung – so berücksichtigt die Gesamtzahl der Dateien nur die Anzahl der insgesamt enthaltenen komprimierten Dubletten von 0.dll und nicht die ebenfalls mit eingebetteten Containerdateien (deren Anteil an der Gesamtmenge allerdings auch vernachlässigbar gering ausfällt).

• ²⁾ Die Datei 42.zip fasst lediglich die 16 Containerdateien der nächst-unteren Hierarchiestufe zu einer einzigen Datei zusammen, daher ändert sich an Gesamtgröße und Anzahl im Vergleich zur nächst-unteren Stufe nichts.

• ³⁾ Die o.a. Notierung der Dateinamen erfolgt nach dem Schema Präfix# — „#“ ist dabei der Platzhalter für eine einstellige Hexadezimalzahl, kann also insgesamt 16 Werte (0, 1, 2, […], 9, A, B, C, D, E, F) annehmen. „lib0.zip … libf.zip“ steht also für 16 durchnummerierte einzelne Dateien.

• ⁴⁾ Plik 0.dll zawiera ciąg 0xAA tworzący 4.294.967.295 B danych, przez to bardzo dobrze poddaje się kompresji bezstratnej.

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Third tunnel of aggression

Trzeci tunel agresji jest tunelem pod granicą między Koreą Północną a Południową wydrążony na południe od Panmundżonu. Jest to trzeci odkryty tunel biegnący pod granicą państw koreańskich.

Historia

Biegnący 44 km od Seulu tunel został odkryty w październiku 1978 r. na podstawie informacji udzielonych przez uciekiniera z Korei Północnej. Tunel ma 1.7 km długości, 2 m wysokości i 2 m szerokości. Biegnie w skale na głębokości 73 m. Wydaje się, że tunel został pomyślany do niespodziewanego ataku na Seul z Korei Północnej. Może nim przejść 30000 żołnierzy z lekkim uzbrojeniem w ciągu godziny.

Upon discovery of the third tunnel, the United Nations Command accused North Korea of threatening the 1953 armistice agreement signed at the end of the Korean War. Its description as a "tunnel of aggression" was given by the South, who considered it an act of aggression on the part of the North.

A total of four tunnels have been discovered so far, but there are believed to be up to ten more. South Korean and U.S. soldiers regularly drill in the Korean Demilitarized Zone in hopes of finding more.

Entrance to the visitor access tunnel

Initially, North Korea denied building the tunnel. However, observed drill marks for dynamite in the walls point towards South Korea and the tunnel is inclined so that water drains back towards the northern side of the DMZ (and thus out of the way of continued excavation).

North Korea then officially declared it part of a coal mine; black "coal" was painted on the walls by retreating soldiers to help confirm this statement. However, statements in the tunnel claim that there is no geological likelihood of coal being in the area. The walls of the tunnel where tourists are taken are observably granite, a stone of igneous origin, whereas coal would be found in stone of sedimentary origin.

Photos are forbidden within the tunnel, which is now well guarded, though it is a busy tourist site, where visitors enter by going down a long steep incline that starts in a lobby with a gift shop. The South Koreans have blocked the actual Military Demarcation Line in the tunnel with three concrete barricades. The third is visible by tourists visiting the tunnel and the second is visible through a window in the third.

See also

• Korean Demilitarized Zone
• Korean People's Army
• List of Korea-related topics

Jugosławia

Xpicto/brudnopis

Československo

Název

Název Česká a Slovenská Federativní Republika byl zaveden 23. dubna 1990 po tzv. „pomlčkové válce“ namísto sporných názvů Československá federativní republika (platný od 29. března 1990 do 22. dubna 1990) a Česko-slovenská federatívna republika. Název je pravopisně nesprávný a vznikl kvůli dalšímu ze sporů tzv. „pomlčkové války“, sporu o velké „S“. Část slovenských představitelů po listopadu 1989 požadovala také velké písmeno „S“ u „Slovenská“, ale zejména čeští zástupci nechtěli na tuto verzi přistoupit.

Tato část sporu se tehdy vedla o označení „Česká a Slovenská federativní republika“ vs. „Česká a slovenská federativní republika“. Podle českého i slovenského pravopisu se přídavná jména vytvořená z vlastních jmen píší malým písmenem. Zároveň se u víceslovných názvů píše velké písmeno pouze u prvního slova (pokud další slova neobsahují jiné vlastní jméno). Jako „šalamounský“ kompromis byl nakonec název „Česká a Slovenská Federativní Republika“ schválen Federálním shromážděním jako ústavní zákon o změně názvu Československé federativní republiky, i když odporoval pravopisu obou jazyků.

„Tehdy se Slováci ohradili, proč mají být Češi s velkým a Slováci s malým.“
–Jiří Pospíšil (ODS)

Pomlčková válka

Pomlčkovou válkou je nazýván spor o název Československa, probíhající po listopadu 1989.

Tento spor, jako i spory o zkratku ČSFR a označení Česko-Slovensko, doprovázely základní spor o osud federace ČSSR, který trval od dob před vznikem Česko-Slovenska, později Československa, v roce 1918.

Nakonec tehdy dostalo Československo název „Česká a Slovenská Federativní Republika“ místo sporných názvů „Československá federativní republika“ a „Česko-slovenská federativní republika“.

Podstata sporu nebyla ani touto formalitou vyřešena a polistopadové Československo/Česko-Slovensko nakonec zaniklo k 31. prosinci 1992, bylo rozděleno na dva samostatné státy – Českou a Slovenskou republiku – formálně založené k 1. lednu 1993. Tento spor byl ještě vyostřen připomenutím názvu Česko-Slovensko, státního útvaru, tzv. „druhé republiky“, která trvala od září 1938 (Mnichovská dohoda a odstoupení pohraničí) do 14. března 1939, kdy Slováci vyhlásili tzv. Slovenskou republiku, 15. března byly české země obsazeny německou armádou a 16. března byl vytvořen Protektorát Čechy a Morava.

Z typografického hlediska je označení pomlčková válka nepřesné, neboť se jednalo o spojovník.

Podle slovenské reformy pravopisu z roku 1991 se ve všech názvech obsahujících „Československo“ nebo „československý“ píše spojovník, tedy jako „Česko-Slovensko“, resp. „česko-slovenský“. Toto pravidlo se v současné době uplatňuje i na starší názvy, které podle něj nebyly vytvořeny.

Serie podręczników do nauki języków obcych

Serie wydawana przez wydawnictwo Wiedza Powszechna. Książki utrzymane są w podobnym stylu, zarówno pod względem graficznym, jak i metodycznym. Zasadniczo książki przeznaczone są dla samouków, posiadają klucz do ćwiczeń. Układ lekcji jest typowy: tekst, słownictwo, objaśnienia gramatyczne, ćwiczenia. Wszystkie serie wygasły na początku lat dziewięćdziesiątych XX wieku.

Uczymy się języków obcych

Logo serii jest biała gwiazda ośmioramienna z nazwą serii wewnątrz. Każdemu językowi odpowiada wiodący kolor.

• Eugeniusz Mroczko, Język węgierski dla początkujących, kolor fioletowy
• Irena Dobrzycka, Bronisław Kopczyński, Język angielski dla początkujących, kolor różowy
• Oskar Perlin, Język hiszpański dla początkujących, kolor niebieski
• Oskar Perlin, Język hiszpański dla zaawansowanych, kolor niebieski
• Maria Szypowska, Język francuski dla początkujących, kolor niebieski
• Maria Łozińska, Ludomir Przestaszewski, Język francuski dla zaawansowanych, kolor niebieski

Poliglota

Logo serii jest dwa koncentryczne kręgi wokół tytułu. Książki były wydawane z płytami gramofonowymi. Każdemu językowi odpowiada wiodący kolor.

• Kazimierz Sabik, Mówimy po hiszpańsku, kolor fioletowy
• Lisetta Stembor, Stanisław Prędota, Mówimy po niderlandzku, kolor oliwkowy
• Danuta Wasilewska, Stanisław Karolak, Mieczysław Jaworowski, Mówimy po rosyjsku, kolor zielony
• Alina Kreisberg, Silvano de Fanti, Mówimy po włosku, kolor niebieski
• A. Płatkow, Mieczysław Jaworowski, Mówimy po francusku, kolor niebieski

Biblioteka Lingwisty

Logo serii są ułożone pionowo litery B, L otoczone nazwą serii. Każdemu językowi odpowiada wiodący kolor.

• Mieczysław Jaworowski, Vorbiţi româneṣte? Zwięzły kurs języka rumuńskiego, kolor zielony
• Maria Krukowska, Govorite li srpskohrvatski? Zwięzły kurs języka serbsko-chorwackiego, kolor zielony
• Sabina Radewa, Говорите ли български? Zwięzły kurs języka bułgarskiego, kolor niebieski

Grafika

Kości nadgarstka:A -- kość łódeczkowata, B -- kość księżycowata, C -- kość trójgraniasta, D -- kość grochowata, E -- kość czworoboczna większa, F -- kość czworoboczna mniejsza, G -- kość główkowata, H -- kość haczykowata

Limba română ca limbă secundară în estul Europei

Limba română ca limbă secundară în estul Europei

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