 6.7.1: 13 compute the SVD of a square singular matrix A.
 6.7.2: 13 compute the SVD of a square singular matrix A.
 6.7.3: 13 compute the SVD of a square singular matrix A.
 6.7.4: 47 ask for the SVD of matrices of rank 2.
 6.7.5: 47 ask for the SVD of matrices of rank 2.
 6.7.6: 47 ask for the SVD of matrices of rank 2.
 6.7.7: 47 ask for the SVD of matrices of rank 2.
 6.7.8: A square invertible matrix has AI = V :E1 UT This says that the s...
 6.7.9: Suppose Ul, ... ,Un and Vb .. . ,Vn are orthononnal bases for Rn. ,...
 6.7.10: Construct the matrix with rank one that has Av 12u for v = !(1, 1, ...
 6.7.11: Suppose A has orthogonal columns WI, W2, .. . ,Wn of lengths (Jl, (...
 6.7.12: Suppose A is a 2 by 2 symmetric matrix with unit eigenvectors UI an...
 6.7.13: If A = QR with an orthogonal matrix Q, the SVD of A is almost the s...
 6.7.14: Suppose A is invertible (with (Jl > (J2 > 0). Change A by as small ...
 6.7.15: Why doesn't the SVD for A + I just use 2: + I?
 6.7.16: Run a random walk x (2), ... , x (n) starting from web site x (1) =...
 6.7.17: The 1, 1 first difference matrix A has AT A = second difference ma...
Solutions for Chapter 6.7: Singular Value Decomposition (SVD)
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 6.7: Singular Value Decomposition (SVD)
Get Full SolutionsChapter 6.7: Singular Value Decomposition (SVD) includes 17 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Since 17 problems in chapter 6.7: Singular Value Decomposition (SVD) have been answered, more than 11106 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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.