 7.4.1E: Find the singular values of the matrices in Exercises 1–4.
 7.4.2E: Find the singular values of the matrices in Exercises
 7.4.3E: Find the singular values of the matrices in Exercises
 7.4.4E: Find the singular values of the matrices in Exercises
 7.4.5E: Find an SVD of each matrix in Exercises. [Hint: In Exercise, one ch...
 7.4.6E: Find an SVD of each matrix in Exercises. [Hint: In Exercise, one ch...
 7.4.7E: Find an SVD of each matrix in Exercises 5–12.
 7.4.8E: Find an SVD of each matrix in Exercises. [Hint: In Exercise, one ch...
 7.4.9E: Find an SVD of each matrix in Exercises. [Hint: In Exercise, one ch...
 7.4.10E: Find an SVD of each matrix in Exercises. [Hint: In Exercise, one ch...
 7.4.11E: Find an SVD of each matrix in Exercises 5–12.[Hint: In Exercise 11,...
 7.4.12E: Find an SVD of each matrix in Exercises 5–12.[Hint: In Exercise 11,...
 7.4.13E: Find the SVD of [Hint: Work with AT]
 7.4.14E: In Exercise 7, find a unit vector x at which Ax has maximum length....
 7.4.15E: Suppose the factorization below is an SVD of a matrix A, with the e...
 7.4.16E: Repeat Exercise 15 for the following SVD of a 3 × 4 matrix A: Refer...
 7.4.17E: In Exercises 17–24, A is an m × n matrix with a singular value deco...
 7.4.18E: In Exercises 17–24, A is an m × n matrix with a singular value deco...
 7.4.19E: In Exercises 17–24, A is an m × n matrix with a singular value deco...
 7.4.20E: In Exercises 17–24, A is an m × n matrix with a singular value deco...
 7.4.21E: In Exercises 17–24, A is an m × n matrix with a singular value deco...
 7.4.22E: In Exercises 17–24, A is an m × n matrix with a singular value deco...
 7.4.23E: In Exercises 17–24, A is an m × n matrix with a singular value deco...
 7.4.24E: In Exercises 17–24, A is an m × n matrix with a singular value deco...
 7.4.25E: Let be a linear transformation. Describe how to find a basis B for ...
 7.4.26E: [M] Compute an SVD of each matrix in Exercises 26 and 27. Report th...
 7.4.27E: [M] Compute an SVD of each matrix in Exercises 26 and 27. Report th...
 7.4.28E: [M] Compute the singular values of the 4 × 4 matrix in Exercise 9 i...
 7.4.29E: [M] Compute the singular values of the 5 × 5 matrix in Exercise 10 ...
Solutions for Chapter 7.4: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 7.4
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since 29 problems in chapter 7.4 have been answered, more than 46662 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.4 includes 29 full stepbystep solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.