A Spectral Blakley–Roy Inequality for Nonnegative Matrices
Let $A$ and $B$ be real symmetric matrices with nonnegative entries. We prove that $\rho(AB)^k\le\rho(A^kB^k)$ for the spectral radius $\rho(\cdot)$ and every integer $k\ge1$; for $B=uu^T$ this is the Blakley–Roy inequality. We prove it via the more general inequality $\|(AB)^m\|^{k}\le\|(A^kB^k)^m\|$ in the spectral norm, valid for all integers $k,m\ge1$. The even exponents case follows from standard tools; for odd exponents $k=2r+1$ the claim follows via the more general product inequality $(u^TC_1\cdots C_tv)^k\le(u^Tu)^r(v^Tv)^r\,u^TC_1^k\cdots C_t^kv$ for nonnegative symmetric $C_i$ and…
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