 9.6.1: Determine the singular values ofthe following matrices. a. A  2 1 ...
 9.6.2: Determine the singular values ofthe following matrices. a. 2 1 I I ...
 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)', (1, 1...
 9.6.6: Given the data 8. Xj 1.0 2.0 3.0 4.0 5.0 y; 1.3 3.5 4.2 5.0 7.0 a. ...
 9.6.7: Given the data Xi yi 1.0 1.84 1.1 1.96 1.3 2.21 1.5 2.45 1.9 2.94 2...
 9.6.8: To determine a relationship between the number of fish and the numb...
 9.6.9: The following set of data, presented to the Senate Antitrust Subcom...
 9.6.10: Suppose that A is an m x n matrix A. Show that Rank(A) is the same ...
 9.6.11: Show that Nullity(A) = Nullity(A') if and only if A is a square mat...
 9.6.12: Suppose that A has the Singular Value Decomposition A U S V. Determ...
 9.6.13: Suppose that A has the Singular Value Decomposition A U S V. Show t...
 9.6.14: Suppose that the m x n matrix A has the Singular Value Decompositio...
 9.6.15: Suppose that the n x n matrix A has the Singular Value Decompositio...
 9.6.16: Part (ii) of Theorem 9.26 states that Nullity(A) = Nullity(A'A). Is...
 9.6.17: Part (iii) of Theorem 9.26 states that Rank(A) Rank*A'A). Is it als...
 9.6.18: Show that if A is an m x n matrix and P is an n x n orthogonal matr...
 9.6.19: Show that if A is an n x n nonsingular matrix with singular values ...
 9.6.20: Use the result in Exercise 19 to determine the condition numbers of...
Solutions for Chapter 9.6: Singular Value Decomposition
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 9.6: Singular Value Decomposition
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Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Column space C (A) =
space of all combinations of the columns of A.

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.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.