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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q33E (page 401)

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

Short Answer

Expert verified

The Cholesky factorization is

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

Step by step solution

01

Given Information:

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

02

Estimating the Cholesky factorization:

We can deduce from Exercise 32 that $2 脳 2$ 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:

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

03

Determining the Result:

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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Most popular questions from this chapter

Let be a complex $n 脳 n$ matrix such that $| 位 | < 1$ for all eigenvalues $位$ of . Show that role="math" localid="1659610526426" $\underset{t 鈫� 鈭�}{l i m} A^{t} = 0$ , meaning that the modulus of all entries of $A^{t}$ approaches zero.

b. Prove Theorem 7.6.2.

Determine the definiteness of the quadratic forms in Exercises 4 through 7.

6. $q (x_{1}, x_{2}) = 2 x_{1}^{2} + 6 x_{1} x_{2} + 4 x_{2}^{2}$

Consider a singular value decomposition $A = U 危 V^{T}$ of an $n 脳 m$ matrix Awith rank $A = m$ . Let ${\overset{鈫�}{u}}_{1}, 鈥�, {\overset{鈫�}{u}}_{n}$ be the columns of U. Without using the results of Chapter 5 , compute $A {(A^{T} A)}^{- 1} A^{T} {\overset{鈫�}{u}}_{i} .$ Explain your result in terms of Theorem 5.4.7.

Consider a symmetric nxnmatrix A with $A^{2} = A$ . Is the linear transformation $T (\overset{鈬赌}{x}) = A \overset{鈬赌}{x}$ necessarily the orthogonal projection onto a subspace of ${鈩�}^{n}$ ?

Consider a symmetric 3x3matrix Awith eigenvalues 1,2and 3how many different orthogonal matricessare there such that $S^{- 1} AS$ is diagonal?

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