- `6.10.1E: Label the following statements as true or false.(a) If Ax = b is we...
- `6.10.2E: Compute the norms of the following matrices.
- `6.10.3E: Prove that if B is symmetric, then ||B|| is the largest eigenvalue ...
- `6.10.4E: Let A and A?1 be as follows: The eigenvalues of A are approximately...
- `6.10.5E: Suppose that x is the actual solution of Ax = b and that a computer...
- `6.10.6E: Let
- `6.10.7E: Let B be a symmetric matrix. Prove that equals the smallest eigenva...
- `6.10.8E: Prove that if ? is an eigenvalue of AA?, then ? is an eigenvalue of...
- `6.10.9E: Prove that if A is an invertible matrix and Ax = b, then
- `6.10.10E: Prove the left inequality of (a) in Theorem 6.44.Theorem 6.44. For ...
- `6.10.11E: Prove that cond(A) = 1 if and only if A is a scalar multiple of a u...
- `6.10.12E: (a) Let A and B be square matrices that are unitarily equivalent. P...
- `6.10.13E: Let A be an n × n matrix of rank r with the nonzero singular values...
Solutions for Chapter `6.10: Linear Algebra 4th Edition
Full solutions for Linear Algebra | 4th Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
peA) = det(A - AI) has peA) = zero matrix.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
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.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
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.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Constant down each diagonal = time-invariant (shift-invariant) filter.