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Q33E

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Found in: Page 401

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Find the Cholesky factorization (discussed in Exercise 32) for ${\mathbit{A}}{\mathbf{=}}\left[\begin{array}{cc}8& 2\\ 2& 5\end{array}\right]$

The Cholesky factorization is

$\left[\begin{array}{cc}8& -2\\ -2& 5\end{array}\right]=\left[\begin{array}{cc}2\sqrt{2}& 0\\ -1/\sqrt{2}& 3/\sqrt{2}\end{array}\right]\left[\begin{array}{cc}2\sqrt{2}& -1/\sqrt{2}\\ 0& 3/\sqrt{2}\end{array}\right]$

See the step by step solution

## Step 1: Given Information:

$A=\left[\begin{array}{cc}8& -2\\ -2& 5\end{array}\right]$

## Step 2: Estimating the Cholesky factorization:

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

$\mathrm{A}=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{b}& \mathrm{c}\end{array}\right]$

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

$\mathrm{L}=\left[\begin{array}{cc}\sqrt{\mathrm{a}}& 0\\ \mathrm{b}/\sqrt{\mathrm{a}}& \sqrt{\mathrm{ac}-{\mathrm{b}}^{2}}/\sqrt{\mathrm{a}}\end{array}\right]$

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

$A=\left[\begin{array}{cc}8& -2\\ -2& 5\end{array}\right]$

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

$L=\left[\begin{array}{cc}2\sqrt{2}& 0\\ -1/\sqrt{2}& 3/\sqrt{2}\end{array}\right]$.

A's Cholesky factorization is as follows:

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

## Step 3: Determining the Result:

The Cholesky factorization is

$\left[\begin{array}{cc}8& -2\\ -2& 5\end{array}\right]=\left[\begin{array}{cc}2\sqrt{2}& 0\\ -1/\sqrt{2}& 3/\sqrt{2}\end{array}\right]\left[\begin{array}{cc}2\sqrt{2}& -1/\sqrt{2}\\ 0& 3/\sqrt{2}\end{array}\right]$