 9.5.1: In Exercises 14, find the distinct singular values of
 9.5.2: In Exercises 14, find the distinct singular values of
 9.5.3: In Exercises 14, find the distinct singular values of
 9.5.4: In Exercises 14, find the distinct singular values of
 9.5.5: In Exercises 512, find a singular value decomposition of A.
 9.5.6: In Exercises 512, find a singular value decomposition of A.
 9.5.7: In Exercises 512, find a singular value decomposition of A.
 9.5.8: In Exercises 512, find a singular value decomposition of A.
 9.5.9: In Exercises 512, find a singular value decomposition of A.
 9.5.10: In Exercises 512, find a singular value decomposition of A.
 9.5.11: In Exercises 512, find a singular value decomposition of A.
 9.5.12: In Exercises 512, find a singular value decomposition of A.
 9.5.13: Prove: If A is an matrix, then and have the same rank.
 9.5.14: Prove part (d) of Theorem 9.5.1 by using part (a) of the theorem an...
 9.5.15: (a) Prove part (b) of Theorem 9.5.1 by first showing that row is a ...
 9.5.16: Let be a linear transformation whose standard matrix A has the sing...
 9.5.17: Show that the singular values of are the squares of the singular va...
 9.5.18: Show that if is a singular value decomposition of then U orthogonal...
 9.5.a: In parts (a)(g) determine whether the statement is true or false, a...
 9.5.b: In parts (a)(g) determine whether the statement is true or false, a...
 9.5.c: In parts (a)(g) determine whether the statement is true or false, a...
 9.5.d: In parts (a)(g) determine whether the statement is true or false, a...
Solutions for Chapter 9.5: Singular Value Decomposition
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 9.5: Singular Value Decomposition
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. Since 22 problems in chapter 9.5: Singular Value Decomposition have been answered, more than 13725 students have viewed full stepbystep solutions from this chapter. Chapter 9.5: Singular Value Decomposition includes 22 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

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.

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.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.