# lecture17 - Lecture 17 The QR Method* The Power Method and...

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Lecture 17 The QR Method* The Power Method and Inverse Power Method each give us only one ew ev pair. While both of these methods can be modiFed to give more ew ’s and ev ’s, there is a better method for obtaining all the ew ’s called the QR method . This is the basis of all modern ew software, including Matlab , so we summarize it brie±y here. The QR method uses the fact that any square matrix has a QR decomposition . That is, for any A there are matrices Q and R such the A = QR where Q has the property Q 1 = Q and R is upper triangular. A matrix Q with the property that its transpose equals its inverse is called an orthogonal matrix, because its column vectors are mutually orthogonal. The QR method consists of iterating following steps: - Transform A into a tridiagonal matrix H . - decompose H in QR . - multiply Q and R together in reverse order to form a new H . The diagonal of H will converge to the eigenvalues. We details of what makes this method converge are beyond the scope of the this book. However,

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## This note was uploaded on 02/09/2012 for the course MATH 344 taught by Professor Young,t during the Fall '08 term at Ohio University- Athens.

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lecture17 - Lecture 17 The QR Method* The Power Method and...

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