# 2021-04 Product of matrices

For an $$n \times n$$ matrix $$M$$ with real eigenvalues, let $$\lambda(M)$$ be the largest eigenvalue of $$M$$. Prove that for any positive integer $$r$$ and positive semidefinite matrices $$A, B$$,

$[\lambda(A^m B^m)]^{1/m} \leq [\lambda(A^{m+1} B^{m+1})]^{1/(m+1)}.$

GD Star Rating
loading...

## 2 thoughts on “2021-04 Product of matrices”

1. Ji Oon Lee Post author

Thank you for the comment. I removed ‘Hermitian’ in the definition of \lambda as A^n B^n is not necessarily Hermitian. (Nevertheless, all eigenvalues of A^n B^n are real.)

Comments are closed.