- 5.6.1E: Let A be a 2 × 2 matrix with eigenvalues 3 and 1/3 and Let be a sol...
- 5.6.2E: Suppose the eigenvalues of a 3 × 3 matrix A are 3, 4/5, and
- 5.6.3E: In Exercises 3–6, assume that any initial vector x0 has an eigenvec...
- 5.6.4E: In Exercises 3–6, assume that any initial vector x0 has an eigenvec...
- 5.6.5E: In Exercises 3–6, assume that any initial vector x0 has an eigenvec...
- 5.6.6E: In Exercises 3–6, assume that any initial vector x0 has an eigenvec...
- 5.6.7E: Let A have the properties described in Exercise 1.a. Is the origin ...
- 5.6.8E: Determine the nature of the origin (attractor, repeller, or saddle ...
- 5.6.9E: In Exercises 9–14, classify the origin as an attractor, repeller, o...
- 5.6.10E: In Exercises 9–14, classify the origin as an attractor, repeller, o...
- 5.6.11E: In Exercises 9–14, classify the origin as an attractor, repeller, o...
- 5.6.12E: In Exercises 9–14, classify the origin as an attractor, repeller, o...
- 5.6.13E: In Exercises 9–14, classify the origin as an attractor, repeller, o...
- 5.6.14E: In Exercises 9–14, classify the origin as an attractor, repeller, o...
- 5.6.15E: eigenvector for A, and two eigenvalues are .5 and .2. Construct the...
- 5.6.16E: [M] Produce the general solution of the dynamical system when A is ...
- 5.6.17E: Construct a stage-matrix model for an animal species that has two l...
- 5.6.18E: A herd of American buffalo (bison) can be modeled by a stage matrix...
Solutions for Chapter 5.6: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications | 5th Edition
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!
Column space C (A) =
space of all combinations of the columns of A.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
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.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).