I am wondering whether there is any relation between principal eigenvalue of sub matrix and the original matrix.
In fact I am facing a problem which is to select $n$ rows and $n$ columns from the original non-negative matrix to construct a new matrix. The principal eigenvalue of the small matrix selected need to be close to certain constant.
I have totally no idea how to start…
I guess figuring out the relation maybe a good starting point of this problem.
UPDATE:
I set up a conceptual optimization problem, hope this can help on the understanding of my problem.
$\min |\max (xAx')-\lambda^*|$
s.t. $\lVert x\rVert_2=1$
$x_i=[w_iv_i]$
$v_i>0$
$w_i=\{0,1\}$
$\sum_i w_i=n$
$A$ is the original matrix, $n$ is the number of rows/columns I used to construct the small matrix, $\lambda^*$ is the target constant of the principal eigenvalue of small matrix.
What I want to know is the $w$ vector
Best Answer
Likely, the best you're going to get will come from the Cauchy Interlace Theorem that @user32309 referred to. You can see a proof of the result here in the paper "Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices" by Suk-Geun Hwang and it states
Theorem 1 (Cauchy Interlace Theorem). Let $A$ be a Hermitian matrix of order $n$, and let $B$ be a principal submatrix of $A$ of order $n - 1$. If $\lambda_n\leq \lambda_{n-1}\leq \dots\leq \lambda_2\leq \lambda_1$, lists the eigenvalues of $A$ and $\mu_n\leq \mu_{n-1}\leq \dots\leq \mu_3\leq \mu_2$ the eigenvalues of $B$, then $\lambda_n\leq \mu_n\leq\lambda_{n-1}\leq\mu_{n-1}\leq \dots\leq \lambda_2\leq \mu_2\leq\lambda_1$.