Eigenvalues under Hadamard product

Question

Consider $\mathbf{A,B}$ two matrices which are unitary and/or Hermitian, What can we say about eigenvalues of their Hadamard product $(\mathbf{A \circ B})$?

Can we bound the eigenvalues in relation to normal matrix product $(\mathbf{AB})$

Thanks!

Answer 1

Since $\bf{A} \circ B$ is a submatrix of the Kronecker product $\bf{A} \otimes B$, one immediately has that

$$\rho(A \circ B) \le \rho(A \otimes B) = \rho(A) \rho(B)$$

where $\rho$ denotes the spectral radius (maximum, in absolute value, of the eigenvalues) of the matrix.

Also note that the spectrum of $A \otimes B$ are completely specified by all possible products of eigenvalues from $A$ and $B$.

Other things that may be of interest to you are Cauchy's interlacing theorem, Schur product theorem.

The best place to start looking for more is Matrix Theory and Applications By Charles R. Johnson, Chapter 3, as mentioned by Santana Afton in a comment.