Suppose that I have a collection of matrices $\{ A_i \mid A_i \in \mathcal{M}(n;\mathbb{R}) \text{ for } 1 \le i \le k \}$. Suppose further that I know that each $A_i$ has only complex (non-real) roots of its characteristic polynomial (I'm hesitant to use the word eigenvalue, as there is no corresponding (real) eigenvector), furthermore suppose that all roots have modulus less than 1. Now, several things seem like they ought to be obvious, but I don't know how to prove them. First, it seems that if I consider the image of the standard $(n-1)$-sphere under $A_i$ (i.e., $A_i \sim S^{n-1} \subset \mathbb{R}^n$) that the resulting ellipsoid (i.e., the image) lies "mostly" inside the standard $(n-1)$-sphere. How precise can "mostly" be made and how can one prove the precise statement. If the preceding is true, then it would seem that (so long as the matrices are "independent") products of such matrices must be contractions "almost everywhere" (at least if I consider countably infinite products, in which case I would expect the product to be zero almost everywhere). I can't imagine a universe where these statements are not true, but am struggling to make them precise and construct proofs. Any suggestions or pointers would be most welcome.
In response the thoughtful comments below, let me expand a bit:
In my original posting, I was striving for brevity and was, perhaps too terse – I apologize. The situation in which I am interested is random matrices (appropriately scaled by $1/\sqrt{n}$). I am principally interested in the case of very large $n$ (my ultimate interest is asymptotic in $n$), and I want to understand the geometric consequences of the Circular Law (i.e., I am familiar with the result that the scaled eigenvalues are distributed uniformly in the unit disk). I do understand that the convergence to the Circular Law is only in distribution, so that one may have “many” (in the case of finite $n$) “ill-behaved” eigenvalues. Still, it seems to me that knowing that for large $n$, the majority of the eigenvalues will have complex modulus less than one ought to tell me something about the Lipschitz constant (operator norm) of the random matrix over some “reasonably large portion” of the $(n-1)$-sphere.
Ultimately, I am interested in products of random matrices (the $k$ in my original posting). As the eigenvectors of i.i.d. random matrices are distributed uniformly on the (complex) $(n-1)$-sphere, it seems that I ought to be able to say that, for $k$ sufficiently large, the Lipschitz constant of the product is bounded by $\epsilon$ away from a set of measure $\delta$ - a statement of this form is my ultimate goal.
I do know that the eigenvalues of a $k$-fold product of i.i.d. random matrices converges to the $k^{\mbox{th}}$ power of the uniform distribution. Thus, in terms of complex modulus, the eigenvalues quickly concentrate near zero. But, I would prefer to some sort of estimate of the behavior in terms of each matrix in the product. It seems, maybe, that explicit estimates of this form might be more difficult to produce than I originally thought.
