Relationship between eigenvalues of a square matrix and its transpose product

Question

Is there a relationship between the eigenvalues of a (3x3) square matrix $B$ (say it has eigenvalues $0,1,2$) and the eigenvalues of $B^T B$? How about the relationship between the eigenvalues of $B$ to $(B+I)^{-1}$?

Answer 1

We can use the spectral theorem to give us a bit of insights of what's going on, that is

If $B$ is a square matrix of eigenvalues $\lambda_k^B$, then $$\sum\limits_{k=1}^n \vert \lambda_k^B \vert^2 \leq \sum\limits_{k=1}^n \lambda_k^{B^TB} $$ In the extreme case where $B$ is normal ($BB^H = B^H B = I$), we have pairwise equality that is $\vert \lambda_k^B \vert^2 = \lambda_k^{B^TB}$

That said, if your $B$ is normal, then eigenvalues of $B^TB $ are just obtained by squaring those of $B$.