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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 5: Q51E (page 234)

a.Consider the matrix product $Q_{1} = Q_{2} S$ , where both $Q_{1}$ and $Q_{2}$ are n脳mmatrices with orthonormal columns. Show that Sis an orthogonal matrix. Hint: Computelocalid="1659499054761" ${Q_{1}}^{T} = Q_{1} = {(Q_{2} S)}^{T} Q_{2} S$ . Note that
${Q_{1}}^{T} = Q = {Q_{2}}^{T} = Q_{2} = I_{m}$
b.Show that the QRfactorization of an n脳mmatrix Mis unique. Hint: If $Q_{1} R_{1} = Q_{2} R_{2}$ , then $Q_{1} = Q_{2} R_{2} {R_{1}}^{- 1}$ . Now use part (a) and Exercise 50a.

Short Answer

Expert verified

Matrix S is orthogonal.
QR factorization of any matrix is unique.

Step by step solution

01

S is orthogonal.

(a)

Observe the following terms

$Q_{1} = Q_{2} S 鈬� S = {Q_{2}}^{T} Q_{1}$

In order to show that S is orthogonal, we need to show that $S^{T} S = l_{m}$ .Consider

$S^{T} S = {({Q_{2}}^{T} Q_{1})}^{T} ({Q_{2}}^{T} Q_{1}) = ({Q_{1}}^{T} {({Q_{2}}^{T})}^{T}) ({Q_{2}}^{T} Q_{1}) = {Q_{1}}^{T} (Q_{2} {Q_{2}}^{T}) Q_{1} = Q_{1} I_{n} Q_{1} = I_{m}$

Thus, S is orthogonal.

02

Consider for part (b).

b)

Let M an n脳 mmatrix and let $M = Q_{1} R_{1} = Q_{2} R$ be QR factorization of M.

Also, recall that the diagonals entries of the matrices $R_{1}$ and $R_{2}$ are strictly positive. Now, it is written as,

$Q_{1} R_{1} = Q_{2} R_{2} = Q_{1} = Q_{2} {R_{2}}^{- 1}$

From part of this series, $R_{1} {R_{1}}^{- 1}$ is an orthogonal matrix. So, it is found as,

$R_{2} {R_{1}}^{- 1} = I 鈬� R_{1} = R_{2}$

And it gives $Q_{1} = Q_{2}$ .

Hence,

Matrix S is orthogonal matrix.
QR factorization of any matrix is unique.

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Most popular questions from this chapter

If A and B are arbitrary $n^{脳} n$ matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

$A^{T} A$ .

Question: Consider an $n 脳 n$ matrix A. Show that A is an orthogonal matrix if (and only if) A preserve the dot product, meaning that $(A \overset{鈫�}{x}) = . (A \overset{鈫�}{y}) = \overset{鈫�}{x} . \overset{鈫�}{y}$ for allrole="math" localid="1659499729556" $\overset{鈫�}{x}$ and $\overset{鈫�}{y}$ in $R^{n}$ .

Find the angle $胃$ between each of the pairs of vectors $\overset{鈬赌}{u}$ and $\overset{鈬赌}{v}$ in exercises 4 through 6.

4. $\overset{鈬赌}{u} = [\begin{matrix} 1 \\ 1 \end{matrix}], \overset{鈬赌}{v} = [\begin{matrix} 7 \\ 11 \end{matrix}] .$

TRUE OR FALSE?If matrix A is orthogonal, then the matrix $A^{T}$ must be orthogonal as well.

Find the length of each of the vectors $\overset{鈬赌}{v}$ In exercises 1 through 3.

3. $\overset{鈬赌}{v} = [\begin{matrix} 2 \\ 3 \\ 4 \\ 5 \end{matrix}]$

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