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. $\lVert x\rVert_2=1$
$A$ 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