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.

Elementary Linear Algebra (MindTap Course List)
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ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.CM: Cumulative Review
Problem 17CM: Find the sequence of the elementary matrices whose product is the non singular matrix below. [2410]
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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.
Transcribed Image Text: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.
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