A Matrix Trace Inequality
Copyright policy )A Matrix Trace Inequality
The author proves that, if \(A\), \(B\) are positive semidefinite \(m\times m\) matrices, then \(0\leq\text{tr}(AB)^n\leq (\text{tr} A)^n(\text{tr} B)^n\) for every positive integer \(n\). This is deduced from two more elaborate inequalities \[ \begin{aligned} & 0\leq \text{tr}(AB)^{2n}\leq (\text{tr} A)^2(\text{tr }A^2)^{n-1}(\text{tr} B^2)^n,\\ & 0\leq \text{tr}(AB)^{2n+1}\leq (\text{tr }A)(\text{tr }B)(\text{tr }A^2)^n(\text{tr }B^2).\end{aligned} \] .
- Tsinghua University China (People's Republic of)
Miscellaneous inequalities involving matrices, Applied Mathematics, matrix trace, trace inequality, Hermitian, skew-Hermitian, and related matrices, Determinants, permanents, traces, other special matrix functions, Hermitian matrix, positive semidefinite, Analysis
Miscellaneous inequalities involving matrices, Applied Mathematics, matrix trace, trace inequality, Hermitian, skew-Hermitian, and related matrices, Determinants, permanents, traces, other special matrix functions, Hermitian matrix, positive semidefinite, Analysis
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