Problem of the week

2015-24 Hölder inequality for matrices

Let $$A, B$$ are $$n \times n$$ Hermitian matrices and $$p, q \in [1, \infty]$$ with $$\frac{1}{p} + \frac{1}{q} = 1$$. Prove that $| Tr (AB) | \leq \| A \|_{S^p} \| B \|_{S^q}.$ (Here, $$\| A \|_{S^p}$$ is the $$p$$-Schatten norm of $$A$$, defined by $\| A \|_{S^p} = \left( \sum_{i=1}^n |\lambda_i|^p \right)^{1/p},$ where $$\lambda_1, \lambda_2, \dots, \lambda_n$$ are the eigenvalues of $$A$$.)