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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.
AzJ
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    you want Horn and Johnson, mentioned at the other question. It is not Hamiltonian matrices involved, it is Hermitian. It is always possible that someone has defined something called Hamiltonian matrix, but news to me. – Will Jagy May 02 '18 at 20:34
  • Sorry it's been a long day. Do you have an idea of which section I should look in? I found nothing non-obvious in the eigenvalue section of the text. – AzJ May 02 '18 at 20:48
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    I don't really know, and perhaps you should get some sleep. user1551 from the other question would know references, but i don't know whether he is awake, i.e. what time zone he is in. – Will Jagy May 02 '18 at 20:57
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    @WillJagy FYI: A real matrix H is called "Hamiltonian" when JH is symmetric, where J is the symplectic form. It is indeed quite distinct from "Hermitian" matrices (which, confusingly, are used to represent Hamiltonian operators in quantum systems). – Jess Riedel May 26 '23 at 22:11

1 Answers1

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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]).
Pietro Paparella
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