Consider a diagonalizable square matrix $A$ of size $N-\text{by}-N$. Now I want to construct another matrix $B = T^T \cdot A \cdot T$, where $T$ is an orthogonal matrix of size $N-\text{by}-M$ (where $M < N$), which obeys $T^T \cdot T = I$, so $B$ is a square $M-\text{by}-M$ matrix.
Provided that I know the spectral decomposition of $A = S_{A}^{\:} \cdot D_{A}^{\:} \cdot S^{-1}_{A}$, can I say something about the spectrum of $B$? Or literally nothing?