- 7.1: Set A = round(10 rand(6)) s = ones(6, 1) b = A s The solution of th...
- 7.2: If a vector y Rn is used to construct an n n Vandermonde matrix V, ...
- 7.3: Construct a matrix C as follows: Set A = round(100 rand(4)) L = tri...
- 7.4: The n n Hilbert matrix H is defined by h(i, j ) = 1/(i + j 1) i, j ...
- 7.5: Use MATLAB to compute the eigenvalues and eigenvectors of a random ...
- 7.6: Set A = round(10 rand(5)); A = A + A_ [X, D] = eig(A) Compute cond(...
- 7.7: Set A = magic(4) and t = trace(A). The scalar t should be an eigenv...
- 7.8: Set A = diag(10 : 1 : 1) + 10 diag(ones(1, 9), 1) [X, D] = eig(A) C...
- 7.9: Construct a matrix A as follows: A = diag(11 : 1 : 1,1); for j = 0 ...
- 7.10: (a) In MATLAB, the simplest way to compute a Householder matrix tha...
- 7.11: Set x1 = (1 : 5) _; x2 = [1, 3, 4, 5, 9]_; x = [x1; x2] Construct a...
- 7.12: To plot y = sin(x), we must define vectors of x and y values and th...
- 7.13: Let A = 4 5 2 4 5 2 0 3 6 0 3 6 Enter the matrix A in MATLAB and co...
- 7.14: Set A = round(10 rand(10, 5)) and s = svd(A) (a) Use MATLAB to comp...
- 7.15: Set A = rand(8, 4) rand(4, 6), [U, D, V ] = svd(A) (a) What is the ...
- 7.16: With each A Rnn, we can associate n closed circular disks in the co...
- 7.17: We can generate a random symmetric 10 10 matrix by setting A = rand...
- 7.18: A nonsymmetric matrix A may have complex eigenvalues. We can determ...
- 7.19: (a) Generate 100 random 55 matrices and compute the condition numbe...
Solutions for Chapter 7: Numerical Linear Algebra
Full solutions for Linear Algebra with Applications | 8th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
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!
peA) = det(A - AI) has peA) = zero matrix.
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.
Remove row i and column j; multiply the determinant by (-I)i + j •
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Constant down each diagonal = time-invariant (shift-invariant) filter.