1
$\begingroup$

Suppose we have $n \times n$ symmetric matrix $A$ with all diagonal entries zero and remaining entries are non negative. Let $B$ be the matrix obtained from $A$ by deleting its $k$th row and $k$th column and remaining entries of $B$ are less equal the corresponding entries of $A$. Then the largest eigenvalue of $B$ is less equal the largest eigenvalue of $A$ and the smallest eigenvalue of $A$ is less equal the smallest eigenvalue of $B$.

$\endgroup$
1
$\begingroup$

It is a standard result in the theory of non-negative matrices:

If $A,B\in\mathbb{R}^{n\times n}$ are such that $0\leq|B|\leq A$, then $\rho(B)\leq\rho(A)$.

See, e.g., Theorem 2.21 in Varga's book.

Just plug in your $A$ and $B$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.