Consider a Cholesky factorization of the matrix appearing in the normal equations for A E R^mxn, A^T ALL^T. Show that the columns of the matrix AL ^-T are orthonormal. In other words, this matrix is orthogonal in the case m = n. Next, based on this fact, describe how to construct a reduced QR factorization of A using the Cholesky factorization of A^TA.
Consider a Cholesky factorization of the matrix appearing in the normal equations for A E R^mxn, A^T ALL^T. Show that the columns of the matrix AL ^-T are orthonormal. In other words, this matrix is orthogonal in the case m = n. Next, based on this fact, describe how to construct a reduced QR factorization of A using the Cholesky factorization of A^TA.
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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![Consider a Cholesky factorization of the matrix appearing in the normal equations for \( A \in \mathbb{R}^{m \times n} \),
\[ A^T A = L L^T. \]
Show that the columns of the matrix \( A L^{-T} \) are orthonormal. In other words, this matrix is orthogonal in the case \( m = n \). Next, based on this fact, describe how to construct a reduced QR factorization of \( A \) using the Cholesky factorization of \( A^T A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F153bfb9c-b05e-4b91-94e3-9e151aaf7f28%2Ff0e961f4-a459-4bd1-89b5-5cb65d103f46%2F3ug0ses_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a Cholesky factorization of the matrix appearing in the normal equations for \( A \in \mathbb{R}^{m \times n} \),
\[ A^T A = L L^T. \]
Show that the columns of the matrix \( A L^{-T} \) are orthonormal. In other words, this matrix is orthogonal in the case \( m = n \). Next, based on this fact, describe how to construct a reduced QR factorization of \( A \) using the Cholesky factorization of \( A^T A \).
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