$A$ is a matrix that is $k$ x $n$ and is $\in\mathbb{R}$, where n is greater than k. Also, the set of row vectors is orthonormal w.r.t. dot product.
Show $(A^{T}A)^{2}=A^{T}A$.
I know bits and pieces, but can't seem to connect the dots. Here's what I know. Let $\alpha:V$$\rightarrow W$, where V is the vector space defined by $\mathbb{R}^{n}$ and W is the vector space defined by $\mathbb{R}^{k}$ Let B be the basis for V, $B={{v_{1}...v_{n}}}$ and let D be the basis for W, $D={w_{1}...w_{k}}$ Then A is a representation matrix for $\alpha$ and $A^{T}$ is the representation matrix for $\alpha^{*}$, the adjoint operator for $\alpha$. I don't see the connection for a zero dot product making the product matrix squared equal to itself.