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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

Solutions for Chapter 9.6
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ISBN: 9781305253667

This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10. Numerical Analysis was written by and is associated to the ISBN: 9781305253667. Since 20 problems in chapter 9.6: Singular Value Decomposition have been answered, more than 12850 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.6: Singular Value Decomposition includes 20 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• 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 = Fn-l + 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 = M-I 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. A-I is also symmetric.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) 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)j-I and det V = product of (Xk - Xi) for k > i.

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