Let A = 16 -28 0 -28 53 10 0 10 29 a) Prove that A is a positive definite matrix. b) Find the Cholesky factorization A = LLT. b) Using the factorization A = LLT, solve the linear system Ax = b with b = [1 0 1]¹.

Let A = 16 -28 0 -28 53 10 0 10 29 a) Prove that A is a positive definite matrix. b) Find the Cholesky factorization A = LLT. b) Using the factorization A = LLT, solve the linear system Ax = b with b = [1 0 1]¹.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.6: The Algebra Of Matrices

Problem 13E

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Question

Let A =
16 -28 0
-28 53 10
0 10 29
a) Prove that A is a positive definite matrix.
b) Find the Cholesky factorization A = LLT.
b) Using the factorization A = LLT, solve the linear system Ax = b with b = [1 0 1]T.

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Step 1: We give necessary and sufficient condition for positive definite matrix and method of Cholesky.

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Step 2: We prove that matrix A is positive definition matrix.

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Step 3: We find Cholesky factorization.

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Step 4: We apply Cholesky factorization to solve linear systems Ax=b.

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