Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 2 - Matrices and Systems of Linear Equations - 2.7 Elementary Matrices and the LU Factorization - Problems - Page 188: 31

Answer

See below

Work Step by Step

Given any invertible $n \times n$ matrix has a factorization of the form $A = QR$, where $Q$ and $R$ are invertible, $R$ is upper triangular, and $Q$ satisfies $Q^T Q = I_n$. Then the system $Ax=b$ is equivalent to the two problems $$Qy=b\\Rx=y$$ Transfer the first equation to $Q^TQy=y=Q^Tb$ Then, we have $Rx=Q^Tb$. Since $R$ is upper triangular, as a result, we can find x by back substitution
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.