Suppose A is tridiagonal and symmetric in the QR method. From A1 = Q-1 AQ show that A1 is symmetric. Write A1 = RAR-1 to show that A1 is also tridiagonal. (If the lower part of A1 is proved tridiagonal then by symmetry the upper part is too.) Symmetric tridiagonal matrices are the best way to start in the QR method.
Suppose A is tridiagonal and symmetric in the QR method. From A1 = Q-1 AQ show that A1 is symmetric. Write A1 = RAR-1 to show that A1 is also tridiagonal. (If the lower part of A1 is proved tridiagonal then by symmetry the upper part is too.) Symmetric tridiagonal matrices are the best way to start in the QR method.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Suppose A is tridiagonal and symmetric in the QR method. From A1 = Q-1 AQ show that A1 is symmetric. Write A1 = RAR-1 to show that A1 is also tridiagonal. (If the lower part of A1 is proved tridiagonal then by symmetry the upper part is too.)
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As per the question we are given a tridiagonal matrix A whose QR decomposition is symmetric. Next we are given a matrix A1 Q-1AQ and we have to show that A1 is symmetric as well as tridiagonal.
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