Chapter 6.4, Problem 19E

Let A be a $5\times 5$ matrix with real entries. Let $A=QT{Q}^T$ be the real Schur decomposition of A, where T is a block matrix of the form given in equation (2). What are the possible block structures for T in each of the following cases?

1. All of the eigenvalues of A are real.
2. A has three real eigenvalues and two complex eigenvalues.
3. A has one real eigenvalue and four complex eigenvalues.