How to generate square matrices based on constraints on their spectral radii?

Question

After reading the Wikipedia, I am wondering whether there is an algorithm that, given a size $N$, can generate an $N\!\times\! N$ matrix $\boldsymbol{W}$ whose spectral radius satisfies $\rho(\boldsymbol{W})<1$, where

$$ \rho(\boldsymbol{W}) = \max\limits_{\boldsymbol{W}} \left\{ |\lambda_{1}|, \dots, |\lambda_{n}| \right\} $$

for some natural $n$.

Answer 1

Here is a simple (albeit inefficient) algorithm to do this.

1. Randomly generate a matrix $W_0$ (without a constraint on its spectral radius)
2. Compute $r_0 = \rho(W_0)$
3. Generate a random value $r$ with $0 \leq r < 1$
4. Obtain the desired matrix with $W = \frac{r}{r_0}W_0$. We have $\rho(W) = r < 1$.

We can make this process faster if we merely require that $r_0 \geq W_0$, which means that that the resulting $W$ has spectral radius $\rho(W) \leq r$. Towards that end, we can simply take $r_0 = \|W\|$ for any submultiplicative norm $\|\cdot\|$. Note, however, that this makes it relatively unlikely that $\rho(W)$ will be close to $1$.