Editing Orthogonal matrix (section)

==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 variously called "semi-orthogonal matrices", "orthonormal matrices", "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".

For the case {{math|''n'' ≤ ''m''}}, matrices with orthonormal columns may be referred to as [[k-frame|orthogonal k-frames]] and they are elements of the [[Stiefel manifold]].