# A trace inequality for positive definite matrices

Journal Article

### Source:

Journal of Inequalities in Pure and Applied Mathematics (JIPAM), Volume 10, Number 1, p.1-4 (2009)

### Keywords:

positive definite matrices, positive semidefinite matrices, trace inequality

### Abstract:

In this note we prove that $Tr {MN + PQ} ≥ 0$ when the following two conditions are met: (i) the matrices $M, N, P, Q$ are structured as follows $M = A − B, N = B^{−1} − A^{−1}$, $P = C − D, Q = (B + D)^{−1} − (A + C)^{−1}$ (ii) $A, B$ are positive definite matrices and $C, D$ are positive semidefinite matrices.

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