Let $A \in M_{n,k}(\mathbb{R})$ such that : $^{t\mkern-5mu}AA = I_k$ then what can be said about : $A\, ^{t\mkern-5mu}A$ ? In which way should I think about the transformation $A^tA$ ?
First of when $n = k$ it's easy since $A$ is orthogonal so $A$ is basically a rotation or a reflexion. Now it's more difficult when $n \ne k$.
The fact that : $^{t\mkern-5mu}A A =I_k$ means that $k > n$ since otherwise $A$ can't be surjective. So we can deal with the case $k > n$ only. When $k > n$ it means that $A$ is surjective and $^{t\mkern-5mu}A$ is injective. So when looking at : $A\, ^{t\mkern-5mu}A$ we are basically taking a vector $v$ in $\mathbb{R}^n$ and return a vector $a \in \mathbb{R}^k$ such that $Aa = v$ and then apply $A$. So it should give us : $A\,^{t\mkern-5mu}A = I_n$ yet this is false... So here my intuition is flawed.
Thank you !