Given two real matrices ${\bf A}$ and ${\bf B}$ of shape $m \times n$ with $m > n$. Let the columns within each of the matrices ${\bf a}_j$ and ${\bf b}_j$ be orthogonal and of unit norm. Such that $$ {\bf A}^T{\bf A} = {\bf I}\quad\text{and}\quad {\bf A}{\bf A}^T \neq {\bf I}\text{,} $$ $$ {\bf B}^T{\bf B} = {\bf I}\quad\text{and}\quad {\bf B}{\bf B}^T \neq {\bf I}\text{.} $$ Then, define the $n \times n$ matrix ${\bf C} = {\bf A}^T{\bf B}$ with respective eigenvalues $\lambda_i$.
Is there a respective upper and lower bound on the absolute value of the largest $|\lambda_\text{max}|$ and smallest $|\lambda_\text{min}|$ eigenvalue? And more importantly, when are these bounds achieved?
I have the hypotheses that $|\lambda_\text{max}|$ should be smaller $1$ as long as ${\bf A} \neq {\bf B}$ and equal to $1$ if ${\bf A} = {\bf B}$; and that $|\lambda_\text{min}|$ should be larger $0$ as long as none of the columns are reversed, i.e., no ${\bf a}_j = - {\bf b}_j$, but don't know how to show it.
Denote $BA^t$ with $D$. We have $\lambda(A^tB)/0=\lambda(BA^t)/0.$ And $\lambda(D^tD)/0=\lambda(AB^tBA^t)/0=\lambda(AA^t)/0=\lambda(A^tA)/0=1.$ Consider the definition of spectral norm, then $\vert\lambda(C)\vert_\max=\vert\lambda(D)\vert_\max\leq 1.$