Chapter 5.2, Problem 23E

A widely used method for estimating eigenvalues of a general matrix A is the QR algorithm. Under suitable conditions, this algorithm produces a sequence of matrices, all similar to A. that become almost upper triangular, with diagonal entries that approach the eigenvalues of A. The main idea is to factor A (or another matrix similar to A) in the form A = Q₁R₁. where $Q_1^T=Q_1^{-1}$ and R₁ is upper triangular The factors are interchanged to form A₁ = Q₁R₁, which is again factored as A₁ = Q₂R₂. then to form A₂ = Q₂R₂, and so on. The similarity of A, A₁, .... follows from the more general result in Exercise 23.

23. Show’ that if A = QR with Q invertible, then A is similar to A₁ = RQ.