 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 stagematrix 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
ISBN: 9780321982384
Solutions for Chapter 5.6
Get Full SolutionsLinear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.6 includes 18 full stepbystep solutions. Since 18 problems in chapter 5.6 have been answered, more than 47404 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5.

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

Cyclic shift
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 Fn1c 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+ (MoorePenrose 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 = UI.
Orthonormal columns (complex analog of Q).