Math 490 - Spring 2019

Homework 8

Due on Friday, March 22.

  1. Suppose that \(A, B, S \in M_n\) and that \(S^{-1} A S\) and \(S^{-1} B S\) are both upper triangular matrices. Prove that all eigenvalues of \(AB-BA\) must be zero.

  2. Suppose that \(A \in M_n\) is unitarily equivalent to \(B\). Prove that \(A\) and \(B\) have the same Frobenius norm. Recall that the Frobenius norm of a matrix with entries \(a_{ij}\) is \(\sqrt{\sum_{ij} |a_{ij}|^2}\). We also proved that \(\|A\|_{Frobenius} = \sqrt{\operatorname{trace} A^*A}\).

  3. Suppose that \(A \in M_n\) has eigenvalues \(\lambda_1, \ldots, \lambda_n\). Use Schur's Triangularization Theorem to prove that \(A\) is unitarily diagonalizable if and only if \(\operatorname{trace}(A^*A) = \sum_{i = 1}^n |\lambda_i|^2\).

  4. Show that the eigenvalues of \(A = \begin{bmatrix} 2 & -5 \\ 1 & -2 \end{bmatrix}\) are \(\pm i\).

  5. Explain why the matrix \(A\) in the last problem is not unitarily diagonalizable.