For an \( n \times n \) matrix \( M \) with real eigenvalues, let \( \lambda(M) \) be the largest eigenvalue of \( M\). Prove that for any positive integer \( r \) and positive semidefinite matrices \( A, B \),
\[[\lambda(A^m B^m)]^{1/m} \leq [\lambda(A^{m+1} B^{m+1})]^{1/(m+1)}.\]
Let \( A, B \) be \( n \times n \) Hermitian matrices. Find all positive integer \( n \) such that the following statement holds:
“If \( AB – BA \) is singular, then \( A \) and \( B \) have a common eigenvector.”
Let \( A, B, C \) be \( N \times N \) Hermitian matrices with \( C = A+B \). Let \( \alpha_1 \geq \dots \geq \alpha_N \), \( \beta_1 \geq \dots \geq \beta_N \), \( \gamma_1 \geq \dots \geq \gamma_N \) be the eigenvalues of \( A, B, C \), respectively. For any \( 1 \leq k \leq N \), prove that
\[ \gamma_1 + \gamma_2 + \dots + \gamma_k \leq (\alpha_1 + \alpha_2 + \dots + \alpha_k) + (\beta_1 + \beta_2 + \dots + \beta_k) \]
Let \( A, B, C = A+B \) be \( N \times N \) Hermitian matrices. Let \( \alpha_1 \geq \cdots \geq \alpha_N \), \( \beta_1 \geq \cdots \geq \beta_N \), \( \gamma_1 \geq \cdots \geq \gamma_N \) be the eigenvalues of \( A, B, C \), respectively. For any \( 1 \leq i, j \leq N \) with \( i+j -1 \leq N \), prove that
\[ \gamma_{i+j-1} \leq \alpha_i + \beta_j \]