let the matrices $M, N \in \mathbb{R}^{n\times n}$ be positive definite, and $M$ is a lower block diagonal matrix. Consider the matrix $T \in \mathbb{R}^{n\times n}$
$$ T = N^T M N .$$
Can I say something about the boundedness of the eigenvalues of the matrix $T$? I know they are bounded from below by zero, but I'm searching for an upper bound for the eigenvalues. Can I maybe use the eigenvalues of M for that?
This does not look like a new or unstudied problem. Does someone know some references?
Kind regards and thanks in advance!