Simultaneous Contractibility
Tsuyoshi Andô, Mau-Hsiang Shih · SIAM Journal on Matrix Analysis and Applications · 1998
Let C be a set of n x n complex matrices. For m = 1,2,...,\ C m is the set of all products of matrices in C of length m. Denote by $\hat{r}({\cal C})$ the joint spectral radius of C, that is, $$\hat{r}({\cal C})\stackrel{\rm def}{=}\limsup_{m\to\infty}[\sup_{A\in {\cal C}^m}\| A\| ]^{\frac{1}{m}}.$$ We call Csimultaneously contractible if there is an invertible matrix S such that $$\sup\{\| S^{-1}AS\|;\ A\in {\cal C}\}<1,$$ where $\| \cdot\|$ is the spectral norm. This paper is primarily devoted to determining the optimal joint spectral radius range for simultaneous contractibility of bounded sets of n x n complex matrices, that is, the maximum subset J of [0,1) such that if C is a bounded set of n x n complex matrices and $\hat{r}({\cal C})\in J$, then C is simultaneously contractible. The central result proved in this paper is that this maximum subset is $[0,\frac{1}{\sqrt{n}}).$ Our method of proof is based on a matrix-theoretic version of complex John's ellipsoid theorem and the generalized Gelfand spectral radius formula.