A spatial form of the spectral-radius formula for matrices

Abstract:  For an m x m complex matrix A, we will discuss the limiting behaviour of the matrix sequence { |A^n|^(1/n) } and provide a stronger form of Yamamoto's theorem which asserts that s_j(A^n)^(1/n) tends to the jth-largest eigenvalue-modulus of A, where s_j(A^n) denotes the jth-largest singular value of A^n (j=1 gives us the spectral radius formula). This also tells us about the limiting behaviour of the sequence { |g^n|^(1/n)} for an element g in a real non-compact semsimple Lie group with a given Cartan involution. Towards the end, we will discuss connections with the invariant subspace problem and discuss the difficulties in studying the limiting behaviour of such sequences when A is a bounded operator on an infinite-dimensional Hilbert space.

Slide (pdf)