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If a square matrix $\mathbf{A}$ satisfies $\mathbf{A}^{H}\mathbf{A}=\mathbf{I}$, then is it a unitary matrix? Or, is there a counter example with $\mathbf{A}\mathbf{A}^{H}\neq\mathbf{I}$?

vonbrand
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nuse_li
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    If A has a left inverse then it is also the right inverse for any square matrix A. –  Mar 17 '13 at 13:34
  • You can find the proof of the above fact here: http://math.stackexchange.com/questions/172872/short-matrix-algebra-question/172897#172897 –  Mar 17 '13 at 13:35
  • We should add that the left inverse and right inverse are equal (if one exists), which is really the question posed here. – hardmath Mar 17 '13 at 13:53

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When $A^HA=I$ we know that $A^H = A^{-1}$ (from the definition of an inverse). If we multiply on both sides with $A$ we have $$A A^H = A A^{-1}=I$$

Dominic Michaelis
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