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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 QT1=Q−11 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.
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Linear Algebra and Its Applications (5th Edition)
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