91Ó°ÊÓ

Let \(A\) and \(B\) be \(n \times n\) matrices. Show that if none of the eigenvalues of \(A\) are equal to \(1,\) then the matrix equation \\[ X A+B=X \\] will have a unique solution.

Show that the diagonal entries of a Hermitian matrix must be real.

Let \(A\) be a diagonalizable matrix whose eigenvalues are all either 1 or \(-1 .\) Show that \(A^{-1}=A\)

Show that any \(3 \times 3\) matrix of the form \\[ \left(\begin{array}{lll} a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & b \end{array}\right) \\] is defective.

Let \(A\) be an Hermitian matrix and let \(B=i A .\) Show that \(B\) is skew Hermitian.

An \(n \times n\) matrix \(A\) is said to be idempotent if \(A^{2}=A .\) Show that if \(\lambda\) is an eigenvalue of an idempotent matrix, then \(\lambda\) must be either 0 or 1

Show that if \(A\) is symmetric positive definite, then \(\operatorname{det}(A)>0 .\) Give an example of a \(2 \times 2\) matrix with positive determinant that is not positive definite.

Let \(A\) be a \(4 \times 4\) matrix and let \(\lambda\) be an eigenvalue of multiplicity 3. If \(A-\lambda I\) has rank 1 , is \(A\) defective? Explain.

Let \(A\) and \(C\) be matrices in \(\mathbb{C}^{m \times n}\) and let \(B \in \mathbb{C}^{n \times r}\) Prove each of the following rules: (a) \(\left(A^{H}\right)^{H}=A\) (b) \((\alpha A+\beta C)^{H}=\bar{\alpha} A^{H}+\bar{\beta} C^{H}\) (c) \((A B)^{H}=B^{H} A^{H}\)

An \(n \times n\) matrix is said to be nilpotent if \(A^{k}=O\) for some positive integer \(k .\) Show that all eigenvalues of a nilpotent matrix are 0