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












1












$begingroup$


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.






matrices reference-request eigenvalues-eigenvectors







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  • 1




    $begingroup$
    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.
    $endgroup$
    – Will Jagy
    May 2 '18 at 20:34












  • $begingroup$
    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.
    $endgroup$
    – AzJ
    May 2 '18 at 20:48






  • 1




    $begingroup$
    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.
    $endgroup$
    – Will Jagy
    May 2 '18 at 20:57
















1












$begingroup$


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.






matrices reference-request eigenvalues-eigenvectors







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    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.
    $endgroup$
    – Will Jagy
    May 2 '18 at 20:34












  • $begingroup$
    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.
    $endgroup$
    – AzJ
    May 2 '18 at 20:48






  • 1




    $begingroup$
    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.
    $endgroup$
    – Will Jagy
    May 2 '18 at 20:57














1












1








1


1



$begingroup$


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.






matrices reference-request eigenvalues-eigenvectors







share|cite|improve this question











$endgroup$




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.






matrices reference-request eigenvalues-eigenvectors




matrices reference-request eigenvalues-eigenvectors






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited May 2 '18 at 20:42







AzJ

















asked May 2 '18 at 20:26









AzJAzJ

295219




295219








  • 1




    $begingroup$
    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.
    $endgroup$
    – Will Jagy
    May 2 '18 at 20:34












  • $begingroup$
    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.
    $endgroup$
    – AzJ
    May 2 '18 at 20:48






  • 1




    $begingroup$
    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.
    $endgroup$
    – Will Jagy
    May 2 '18 at 20:57














  • 1




    $begingroup$
    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.
    $endgroup$
    – Will Jagy
    May 2 '18 at 20:34












  • $begingroup$
    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.
    $endgroup$
    – AzJ
    May 2 '18 at 20:48






  • 1




    $begingroup$
    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.
    $endgroup$
    – Will Jagy
    May 2 '18 at 20:57








1




1




$begingroup$
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.
$endgroup$
– Will Jagy
May 2 '18 at 20:34






$begingroup$
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.
$endgroup$
– Will Jagy
May 2 '18 at 20:34














$begingroup$
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.
$endgroup$
– AzJ
May 2 '18 at 20:48




$begingroup$
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.
$endgroup$
– AzJ
May 2 '18 at 20:48




1




1




$begingroup$
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.
$endgroup$
– Will Jagy
May 2 '18 at 20:57




$begingroup$
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.
$endgroup$
– Will Jagy
May 2 '18 at 20:57










1 Answer
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2












$begingroup$

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]).






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    $begingroup$

    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]).






    share|cite|improve this answer











    $endgroup$


















      2












      $begingroup$

      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]).






      share|cite|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        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]).






        share|cite|improve this answer











        $endgroup$



        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]).







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 14 '18 at 23:08

























        answered Dec 14 '18 at 21:37









        Pietro PaparellaPietro Paparella

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