On two sibling problems of linear and bilinear maps
Vuong Bui · arXiv (Cornell University) · 2021
We provide a short proof for the convergence of the $n$-th root of $\|A^n\|$ for a complex matrix $A$ by showing $\|A^n\|$ is weakly submultiplicative and then using Fekete's lemma. This is simpler than the algebraic approach of Gelfand's formula. Instead of the spectral radius, the limit is expressed as the infimum of $\sqrt[n]{D\|A^n\|}$ where $D\times D$ is the dimension of $A$ and $\|.\|$ is the maximum norm. A corollary is an effective bound of the spectral radius. A simple proof using Perron-Frobenius' theorem for the convergence to the spectral radius for nonnegative matrices is also given. We address a sibling problem: Given a fixed bilinear map $*:\mathbb C^d\times \mathbb C^d\to \mathbb C^d$ and a fixed vector $v\in \mathbb C^d$, we prove that the norm of the sequence $v, v*v, (v*v)*(v*v), ((v*v)*(v*v))*((v*v)*(v*v)), \dots$ is superexponential, that is the $(2^n)$-th root converges. Bounds for this limit are also given. More complex sequences are discussed.