 7.7.1: Find the solution x to the least squares problem, given that A = QR...
 7.7.2: Let A = D E = d1 d2 ... dn e1 e2 ... en and b = b1 b2 . . . b2n Use...
 7.7.3: Let A = 1 0 1 3 1 3 1 0 , b = 4 2 2 2 (a) Use Householder transform...
 7.7.4: Given A = 1 5 1 3 1 11 1 5 and b = 1 1 3 5 (a) Use Algorithm 5.6.1 ...
 7.7.5: Let A = 1 1 0 0 where is a small scalar.(a) Determine the singular ...
 7.7.6: Show that the pseudoinverse A+ satisfies the four Penrose conditions.
 7.7.7: Let B be any matrix that satisfies Penrose conditions 1 and 3, and ...
 7.7.8: If x Rm, we can think of x as an m 1 matrix. If x = 0 we can then d...
 7.7.9: Show that if A is a m n matrix of rank n, then A+ = (AT A) 1AT .
 7.7.10: Let A be an mn matrix and let b Rm. Show that b R(A) if and only if...
 7.7.11: Let A be an m n matrix with singular value decomposition UVT , and ...
 7.7.12: Let A = 1 1 1 1 0 0 Determine A+ and verify that A and A+ satisfy t...
 7.7.13: Let A = 1 2 1 2 and b = 6 4 (a) Compute the singular value decompos...
 7.7.14: Show each of the following: (a) (A+) + = A (b) (AA+) 2 = AA+ (c) (A...
 7.7.15: Let A1 = U1VT and A2 = U2VT , where 1 = 1 ... r1 0 ... 0 and 2 = 1 ...
 7.7.16: Let A = XYT , where X is an m r matrix, YT is an r n matrix, and XT...
Solutions for Chapter 7.7: Least Squares Problems
Full solutions for Linear Algebra with Applications  9th Edition
ISBN: 9780321962218
Solutions for Chapter 7.7: Least Squares Problems
Get Full SolutionsChapter 7.7: Least Squares Problems includes 16 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. Since 16 problems in chapter 7.7: Least Squares Problems have been answered, more than 12053 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).