A stronger form of Yamamoto's theorem on singular values

Abstract

For a matrix T∈Mm(C), let |T|:=T⁎T. For A∈Mm(C), we show that the matrix sequence {|An|1n}n∈N converges to a positive-semidefinite matrix H whose jth-largest eigenvalue is equal to the jth-largest eigenvalue-modulus of A (for 1≤j≤m). In fact, we give an explicit description of the spectral projections of H in terms of the eigenspaces of the diagonalizable part of A in its Jordan-Chevalley decomposition. This gives us a stronger form of Yamamoto's theorem which asserts that limn→∞⁡sj(An)1n is equal to the jth-largest eigenvalue-modulus of A, where sj(An) denotes the jth-largest singular value of An. Moreover, we also discuss applications to the asymptotic behaviour of the matrix exponential function, t↦etA.