Hello all,

This is a question that might or might not be related to my previous one.

Imagine you have two matrices:

1. Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M}$ where $M\leq L$. The column-vectors have unit norm $||\Phi_i||=1$, and the maximum dot-product between distinc vectors is $|\Phi^\top_i\Phi_j|\leq\alpha$ for any $i\neq j$.

2. The second matrix $\mathbf{T}\in\mathbb{R}^{L\times L}$ is totally known. It is positive-definite and its eigenvalues have multiplicity 1, i.e. $\lambda_1 > \ldots > \lambda_L > 0$.

My question is... what are all the possible diagonal values of the matrix product $\mathbf{\Phi}^\top\mathbf{T}\mathbf{\Phi}$?

I actually know the solution by the case $\alpha=0$:

In such case $\mathbf{\Phi}$ is simply an orthogonal matrix. Call $\mathbf{V}$ and $\mathbf{\Lambda}$ the eigenvector and eigenvalue matrix of $\mathbf{T}$ respectively. Thus $\mathbf{\Phi}^\top\mathbf{T}\mathbf{\Phi} = \left(\mathbf{\Phi}^\top\mathbf{V}\right)\mathbf{\Lambda}\left(\mathbf{V}^\top\mathbf{\Phi}\right)$ where $\mathbf{\Phi}^\top\mathbf{V}$ is another "orthogonal" matrix (rows are orthogonal). Thus the original question becomes "what diagonal values are possible with a matrix with prescribed eigenvalues?"

This is completely answered by the Schur-Horn theorem and relies on some majortization theory.

Thank you!