Question

Question: 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 \(A = {Q_1}{R_1}\), where \({Q_1}^T = {Q_1}^{ - 1}\) and \({R_1}\) is upper triangular. The factors are interchanged to form \({A_1} = {R_1}{Q_1}\), which is again factored to \({A_1} = {R_{\bf{1}}}{Q_{\bf{1}}}\); then to form \({A_2} = {R_2}{Q_2}\), and so on. The similarity of \(A,{\rm{ }}{A_1},...\) follows from the more general result in Exercise 23.

23. Show that if \(A = QR\) with \(Q\) invertible, then \(A\) is similar to \({A_1} = RQ\).

Short answer

It is proved that \({A_1}\) is similar to \(A\).

Step-by-step solution

Write the given condition

It is given that \(A = QR\) with \(Q\) invertible.

To show that \(A\) is similar to \({A_1} = RQ\), consider \({A_1} = RQ\).

Show that A is similar to \({A_1} = RQ\)

Rewrite the equation \({A_1} = RQ\) as:

\(\begin{gathered} {A_1} = {Q^{ - 1}}QRQ \\ = {Q^{ - 1}}AQ{\text{ }}\left\{ {\because A = QR} \right\} \\ \end{gathered} \)

Thus, this implies that \({A_1}\) is similar to \(A\).