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 \( A, B \),

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

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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.)

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