Can such an "orthogonal" matrix exist? I know that the definition of an orthogonal matrix is that $A \in \mathbb R^{n \times n}$ is orthogonal if $AA^T = A^T A=I$, no problem with that whatsoever.
My question is this - Why only square matrices?
Is it possible that there is a matrix $A \in \mathbb R^{n \times k}$ such that $AA^T =I_n$ and $A^TA= I_k$?
if so - could you find an example? and if not - why not?
This is not for homework, this is merely my curiosity

Got it. if $AA^T=I_n$ then $A$ has rank $n$ at least, and if $A^TA=I_k$ then $A^T$ has rank $k$ at least. Since $rank(A)=rank(A^T)$ that means $A$ has rank at least $k$ and at least $n$, so they must be equal.
