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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: Q29E (page 400)

For the matrix $A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}],$ write $A = B B^{T}$ as discussed in Exercise 28. See Example 1.

Short Answer

Expert verified

$A = B B^{T} w h e r e B = \frac{1}{\sqrt{5}} [\begin{matrix} - 6 & 2 \\ 3 & 4 \end{matrix}]$

Step by step solution

01

Given Information:

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

02

Estimating S:

Consider the matrix below:

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

Now, as shown below, calculate the eigen values:

$\det (A - 位濒) = 0$

$鈬� |\begin{matrix} 8 - 位 & - 2 \\ - 2 & 5 - 位 \end{matrix}| = 0$

$鈬� 位^{2} - 13 位 + 36 = 0 鈬� (位 - 4) 路 (位 - 9) = 0$

As a result, the eigenvalues are 9 and 4, yielding

$D = [\begin{matrix} 9 & 0 \\ 0 & 4 \end{matrix}]$

So,

$D_{1} = [\begin{matrix} 3 & 0 \\ 0 & 2 \end{matrix}]$

The eigen vectors are as follows:

$[\begin{matrix} - 2 \\ 1 \end{matrix}] and [\begin{matrix} 1 \\ 2 \end{matrix}]$

Find S by combining the eigen vectors after they have been converted to unit vectors, as shown below.

$\frac{1}{\sqrt{5}} [\begin{matrix} - 2 \\ 1' \end{matrix}] a n d \frac{1}{\sqrt{5}} [\begin{matrix} 1 \\ 2 \end{matrix}]$

As a result, we get S, as shown below:

$S = \frac{1}{\sqrt{5}} [\begin{matrix} - 2 & 1 \\ 1 & 2 \end{matrix}]$

03

Estimating B:

Now, as shown below, evaluate B:

$B = {SD}_{1}$

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

$= \frac{1}{\sqrt{5}} [\begin{matrix} - 6 + 0 & 0 + 2 \\ 3 + 0 & 0 + 4 \end{matrix}]$

$鈬� B = \frac{1}{\sqrt{5}} [\begin{matrix} - 6 & 2 \\ 3 & 4 \end{matrix}]$

As a result, we can write A as:

$A = {BB}^{T}$

04

Determining the Result:

$A = B B^{T} w h e r e B = \frac{1}{\sqrt{5}} [\begin{matrix} - 6 & 2 \\ 3 & 4 \end{matrix}]$

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Most popular questions from this chapter

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 the linear transformation $T (q (x_{1} x_{2})) = q (x_{1}, 2 x_{2})$ from $Q_{2} to Q_{2}$ . Find all the eigenvalues and eigenfunctions of . Is transformation diagonalizable?

Let Abe anmatrix and $\overset{鈫�}{v}$ a vector in $R^{m} .$ Show that

$蟽_{m} || \overset{鈫�}{v} || 鈮� || A \overset{鈫�}{v} || 鈮� 蟽_{1} || \overset{鈫�}{v} ||$

where $蟽_{1} a n d 蟽_{m}$ are the largest and the smallest singular values of A, respectively. Compare this with Exercise 25.

If is an indefinite $n 脳 m$ matrix, and R is any real $n 脳 m$ matrix, what can you say about the definiteness of the matrix role="math" localid="1659684209026" $R^{T} AR$ ?

Consider an orthogonal matrix Rwhose first column is $\overset{鈬赌}{v}$ . Form the symmetric matrix $A = \overset{鈬赌}{v} {\overset{鈬赌}{v}}^{T}$ . Find an orthogonal matrix Sand a diagonal matrix Dsuch that $S^{- 1} AS = D$ . Describe Sin terms ofR.

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