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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: Q19E (page 412)

Consider an $n 脳 m$ matrix Aof rank r , and a singular value decomposition $A = U {鈭�}_{}^{} v^{T}$ Explain how you can express the least-squares solutions of a system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ as linear combinations of the columns role="math" localid="1660733836229" ${\overset{鈫�}{v}}_{1} . . . ., {\overset{鈫�}{v}}_{m}$ of V

Short Answer

Expert verified

$x = {鈭�}_{i - 1}^{m} y_{i} U_{i}$

Step by step solution

01

To explaining the least-squares solutions of a system Ax→=b→ as linear combinations of the columns v→1,...,v→m of V

A - $n 脳 m$ matrix

$A = U {鈭� V 路}_{}^{}$ $路路路$ (Singular value decomposition)

Rank $(A) = r$ . Sis are singular values. Denote,

${鈭�}_{}^{(- 1)} = [\begin{matrix} S_{1}^{- 1} \\ S_{r}^{- 1} \\ 0 \\ 0 \end{matrix}]$

Now, x is a least-squares solution to

$A x = b$

if and only if xis a solution of

$A 路 A x = A 路 b .$ Now,

$A 路 A = (V {鈭� U 路}_{}^{}) (U {鈭�}_{}^{} V 路) = V {鈭�^{2}}_{}^{} V^{T}$

02

To explaining the least-squares solutions of a system Ax→=b→ as linear combinations of the columns v→1,...v→m of V

So, we have

$(V {鈭�}_{}^{2} V 路) x = (U {鈭� V 路}_{}^{}) {}^{路 b}b 鈬� V {鈭� V 路}_{}^{2} x = {鈭�}_{}^{} U 路 b (Multiply both sides 鈬� V 路 x = {鈭�}_{}^{(- 1)} U 路 b 鈬� x = V ({鈭�}_{}^{(- 1)} U 路 b) . . . call {鈭�}_{}^{(- 1)} U 路 b = y = [y_{1} . . ., y_{m}] 路, then, {x = 鈭�}_{i - 1}^{m} y_{i} U_{i}$

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Most popular questions from this chapter

Let $J_{n}$ be the nxnmatrix with all ones on the "other diagonal" and zeros elsewhere. (In Exercises 24 and 25, we studied $J_{4}$ and $J_{5}$ , respectively.) Find the eigenvalues of $J_{n}$ , with their multiplicities.

Consider an matrix A, an orthogonal $n 脳 n$ matrix S, and an orthogonal $m 脳 m$ matrix R. Compare the singular values of A with those of SAR.

For which square matrices Ais there a singular value decomposition $A = U 危 V^{T} w i t h U = V$ ?

Let $位$ be a real eigenvalue of an n x n matrix A. Show that

$蟽_{n} 猢� | 位 | 猢� 蟽_{1},$

where $蟽_{1} a n d 蟽_{n}$ are the largest and the smallest singular values of A, respectively.

Show that an SVD $A = U 危 V^{T}$

can be written as $A = 蟽_{1} {\overset{鈫�}{u}}_{1} {\overset{鈫�}{v}}_{1}^{T} + 鈥� + 蟽_{r} {\overset{鈫�}{u}}_{r} {\overset{鈫�}{v}}_{r}^{T} .$

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