Is there any relation between the principal eigenvalue of sub matrix and the original matrix?

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

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There is an interlacing theorem due to Cauchy. –  user32309 May 27 '12 at 11:42