Here's my question --

Let $A$ be an $n \times n$ real matrix, and suppose that the spectral radius $\rho(A)$ is less than one (spectral radius = max eigenvalue). Let's choose some $1 \leq i \leq N$ and look at $A_{N,i}$. Namely, let's replace $A_{N,i}$ with some new value, $a$, to give us a new matrix $\hat A$. I want to characterize the set $\lbrace a : \rho(\hat A) < 1 \rbrace$. It pretty clear that this set is of the form $[0, a_{max})$, but I want to be able to compute $a_{max}$ analytically, given $A$ and $i$. (Also clearly $a_{max} \geq A_{N,i}$, since $\rho(A) < 1$ by assumption.)

This seems like it should be a fairly easy exercise but I haven't been able to make any useful progress on it.

Thanks!

-h

realeigenvalues? If so, the spectral radius will often be $-\infty$ I guess. In any case, I'll give you an example where the spectral radius is increased by replacing an element by 0, which by scaling shows that 0 will not generally be in the set. Let n=N=2 and let A be the matrix with -1 in the bottom right and 1 elsewhere. Let i=2. Then the spectral radius of A is $\sqrt2$, but replacing $A_{22}$ by 0 yields a matrix whose spectral radius is $(1+\sqrt5)/2$. – Jonas Meyer Dec 15 '09 at 5:40