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 \( A \).)

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