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Assume that \(A,B\) are \(n \times n\) matrices.
Carry out each of the following twice: once in the real case and once in the complex case.
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Prove that if \(A,B\) are both product-preserving, then the matrix product \(AB\) is also product-preserving.
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If \(A,B\) are both self-adjoint, is the matrix product \(AB\) necessarily self-adjoint? Prove or find a counter-example.
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Consider the square matrix
\( A = \left[\begin{smallmatrix} A_1 \\ & A_2 \end{smallmatrix}\right] \)
in block-diagonal form.
Carry out each of the following twice: once in the real case and once in the complex case.
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Prove that
\( A^\ast = \left[\begin{smallmatrix} A^\ast_1 \\ & A^\ast_2 \end{smallmatrix}\right] \).
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Prove that if both of the blocks \(A_1,A_2\) are self-adjoint, then so is \(A\).
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Prove that if both of the blocks \(A_1,A_2\) are product-preserving, then so is \(A\).