Maximizing traces of matrix functions
An infinitely differentiable function \(\varphi\colon I\rightarrow \mathbb{R}\) is said to be isoclinically metaconvex on \(I\) if whenever \(t_1,t_2\in I\), \(t_1\neq t_2\), and \(\varphi'(t_1)=\varphi'(t_2)\), then \(\varphi''(t_1)+\varphi''(t_2)>0\). If \(X\) and \(Y\) are two Hermitian \(n\times n\) matrices with eigenvalues \(x_1\geq x_2\geq\cdots\geq x_n\) and, respectively, \(y_1\geq y_2\geq\cdots\geq y_n\), it is proved that \[ \text{tr}(\varphi(X+Y))\leq \max_{\sigma\in S_n}\sum_{j=1}^n \varphi(x_j+y_{\sigma(j)}). \] Also, if \(A\) and \(B\) are positive definite \(n\times n\) matrices with eigenvalues \(a_1\geq a_2\geq \cdots \geq a_n>0\) and, respectively, \(b_1\geq b_2\geq \cdots \geq b_n>0\) and \(t\mapsto \varphi(\text{ e}^t)\) is isoclinically metaconvex, then it is proved that \[ \text{tr}(\phi(AB))\leq \max_{\sigma\in S_n}\sum_{j=1}^n \varphi(a_jb_{\sigma(j)}). \] An application to canonical correlations is also presented.
scientific article; zbMATH DE number 2081246