Suppose that I have a collection of matrices $\{ A_i | A_i \in \mathcal{M}(n;\mathbb{R}) \mbox{ for } 1 \le i \le k \}$. 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., $A_i ~ S^{n-1} \subset \mathbb{R}^n$) 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.