- 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
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 Fn-1c 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.
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 = Q-l. 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(D» 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 A-I are BT AT and (AT)-I.