Are there theorems which relate the eigenvalues of matrices and sub matrices?

Question

I am looking for theorems which relate the eigenvalues square matrices and their submatrices such as https://math.stackexchange.com/a/1670001/374907 but for general matrices not just Hermitian matrices.

I do not know if there exists any theorems on this topic so that I why I am posting this question.

Notes

• Theorems that relate the eigenvalues of $A$, $B$ square matrices and the eigenvalues of $C$ (where $C=A+B$) would be appreciated as well.
• I am not looking for theorems on finding eigenvalues.
• References to books or papers would be appreciated.
• I am not looking for theorems that cover a general square matrices.
• If you need any clarification please feel free to ask.

Answer 1

Two results come to mind:

1. If $A$ is a nonnegative matrix (i.e., $a_{ij} \ge 0$, $1 \le i,j \le n$), then the Perron-Frobenius theorem asserts that the spectral radius $\rho(A)$ is an eigenvalue of $A$. If $\tilde{A}$ is a principal submatrix of $A$, then $\rho(\tilde{A}) \le \rho(A)$ (see, e.g., Corollary 8.1.20 of Matrix Analysis, $2^\text{nd}$ edition, by Horn and Johnson).
2. More recently, the following result was established: let $p$ be a monic polynomial of degree $n$ with roots $\lambda_1,\dots,\lambda_n$ (including multiplicities) and critical points $\mu_1,\dots,\mu_{n-1}$ (including multiplicities). Let $H$ be a complex Hadamard matrix (e.g., $H$ could be the discrete Fourier transform matrix), $D=\text{diag}(\lambda_1,\dots,\lambda_n)$, and $A=HDH^{-1}$. If $A_{(i)}$ denotes the $i$th principal submatrix of $A$, then the characteristic polynomial of $A_{(i)}$ is $p'(t)/n$. Consequently, the eigenvalues of $A_{(i)}$ are $\mu_1,\dots,\mu_{n-1}$ (this result is implied by Theorem 5.12 in Hoover et al. [MR3834205; On the realizability of the critical points of a realizable list. Linear Algebra Appl. 555 (2018), 301–313]).