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Why if the columns of a matrix are orthogonal, then $AA^T\neq A^TA=I$? I know this isn't true if the columns are orthonormal, but since $A^TA=I$, then $A^T=A^{-1}$, and if exists the left inverse, then exists the right inverse such that $AA^{T}=I$. Am I missing something?

Kevin
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    $A$ doesn't have to be square to have orthogonal columns. So $A^TA = I$ doesn't necessarily imply that $A$ is invertible. – Matthew Leingang Jun 12 '19 at 14:29
  • What if we deal only with square matrices? – Kevin Jun 12 '19 at 14:31
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    With orthogonal columns, you only have $A^TA$ diagonal, so you don't have $A^TA=I$. – user10354138 Jun 12 '19 at 14:41
  • The problem arises as I saw this in a question: "Now, if you work out what it means for the columns of $$ to be orthogonal, then that comes out as $^=$. And, the rows of $$ are orthogonal if and only if $^=$" https://math.stackexchange.com/a/2250266/268627 – Kevin Jun 12 '19 at 14:49
  • @JonathanS. I don't see how that conflicts with anything said in the comment section here. If we deal with square matrices, then $A^TA=I$ and $AA^T=I$ are equivalent, and neither can happen without the other. – Arthur Jun 12 '19 at 15:00
  • @Arthur I see, my bad. Thanks! – Kevin Jun 12 '19 at 15:04

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