Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the Cholesky factorization (discussed in Exercise 32) for
$A = [\begin{matrix} 8 & 2 \\ 2 & 5 \end{matrix}]$

The Cholesky factorization is

$[\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}] = [\begin{matrix} 2 \sqrt{2} & 0 \\ - 1 / \sqrt{2} & 3 / \sqrt{2} \end{matrix}] [\begin{matrix} 2 \sqrt{2} & - 1 / \sqrt{2} \\ 0 & 3 / \sqrt{2} \end{matrix}]$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information:

$A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}]$

Step 2: Estimating the Cholesky factorization:

We can deduce from Exercise 32 that $2 \times 2$ positive definite matrix exists.

$A = [\begin{matrix} a & b \\ b & c \end{matrix}]$

The form $A = L L^{T}$ is a Cholesky factorization. Where,

$L = [\begin{matrix} \sqrt{a} & 0 \\ b / \sqrt{a} & \sqrt{ac - b^{2}} / \sqrt{a} \end{matrix}]$

is a positive diagonal matrix with lower triangular entries. The structure of the matrix

$A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}]$

$|A^{(1)}| = 8 > 0$ and $|A^{(2)}| = 40 - 4 = 36 > 0$ are positive definite Thus,

$L = [\begin{matrix} 2 \sqrt{2} & 0 \\ - 1 / \sqrt{2} & 3 / \sqrt{2} \end{matrix}]$ .

A's Cholesky factorization is as follows:

$L L^{T} = [\begin{matrix} 2 \sqrt{2} & 0 \\ - 1 / \sqrt{2} & 3 / \sqrt{2} \end{matrix}] [\begin{matrix} 2 / \sqrt{2} & - 1 / \sqrt{2} \\ 0 & 3 / \sqrt{2} \end{matrix}] = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}] = A$

Step 3: Determining the Result:

The Cholesky factorization is

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Linear Algebra With Applications

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Short Answer

Find the Cholesky factorization (discussed in Exercise 32) for
$A = [\begin{matrix} 8 & 2 \\ 2 & 5 \end{matrix}]$

Step by Step Solution

Step 1: Given Information:

Step 2: Estimating the Cholesky factorization:

Step 3: Determining the Result:

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Linear Algebra With Applications

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Short Answer

Find the Cholesky factorization (discussed in Exercise 32) for A=[8225]

Step by Step Solution

Step 1: Given Information:

Step 2: Estimating the Cholesky factorization:

Step 3: Determining the Result:

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Find the Cholesky factorization (discussed in Exercise 32) for
$A = [\begin{matrix} 8 & 2 \\ 2 & 5 \end{matrix}]$