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 = Q, R1, where Q, =Q,' and R, is upper triangular. The factors are interchanged to form A, =R,Q1, which is again factored as A, = Q2R2; then to form A2 = R2Q2, and so on. The similarity of A, A,,.. follows from the more general result below. Show that if A = QR with Q invertible, then A is similar to A, =RQ.

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