 9.6.1: Determine the singular values of the following matrices. a. A = 2 1...
 9.6.2: Determine the singular values of the following matrices. a. A = 1 1...
 9.6.3: Determine a singular value decomposition for the matrices in Exerci...
 9.6.4: Determine a singular value decomposition for the matrices in Exerci...
 9.6.5: Let A be the matrix given in Example 2. Show that(1, 2, 1)t ,(1, 1,...
 9.6.6: Suppose that A is an m n matrix A. Show that Rank(A) is the same as...
 9.6.7: Show that Nullity(A) = Nullity(At ) if and only if A is a square ma...
 9.6.8: Suppose that A has the singular value decomposition A = USVt . Dete...
 9.6.9: Suppose that A has the singular value decomposition A = USVt . Show...
 9.6.10: Suppose that the m n matrix A has the singular value decomposition ...
 9.6.11: Suppose that the n n matrix A has the singular value decomposition ...
 9.6.12: Part (ii) of Theorem 9.26 states that Nullity(A) = Nullity(At A). I...
 9.6.13: Part (iii) of Theorem 9.26 states that Rank(A) = Rank(At A). Is it ...
 9.6.14: Show that if A is an m n matrix and P is an n n orthogonal matrix, ...
 9.6.15: Show that if A is an n n nonsingular matrix with singular values s1...
 9.6.16: Use the result in Exercise 15 to determine the condition numbers of...
 9.6.17: Given the data xi 1.0 2.0 3.0 4.0 5.0 yi 1.3 3.5 4.2 5.0 7.0 , a. U...
 9.6.18: Given the data xi 1.0 1.1 1.3 1.5 1.9 2.1 yi 1.84 1.96 2.21 2.45 2....
Solutions for Chapter 9.6: Singular Value Decomposition
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 9.6: Singular Value Decomposition
Get Full SolutionsSince 18 problems in chapter 9.6: Singular Value Decomposition have been answered, more than 12754 students have viewed full stepbystep solutions from this chapter. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. Chapter 9.6: Singular Value Decomposition includes 18 full stepbystep solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions.

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Iterative method.
A sequence of steps intended to approach the desired solution.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.