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→Rectangular matrices: Link to the [[semi-orthogonal matrix |
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==Rectangular matrices==
{{Main|Semi-orthogonal matrix}}
If {{mvar|Q}} is not a square matrix, then the conditions {{math|1=''Q''<sup>T</sup>''Q'' = ''I''}} and {{math|1=''QQ''<sup>T</sup> = ''I''}} are not equivalent. The condition {{math|1=''Q''<sup>T</sup>''Q'' = ''I''}} says that the columns of ''Q'' are orthonormal. This can only happen if {{mvar|Q}} is an {{math|''m'' × ''n''}} matrix with {{math|''n'' ≤ ''m''}} (due to linear dependence). Similarly, {{math|1=''QQ''<sup>T</sup> = ''I''}} says that the rows of {{mvar|Q}} are orthonormal, which requires {{math|''n'' ≥ ''m''}}.
There is no standard terminology for these matrices. They are
==See also==
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