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Solutions for Chapter 4.8: Linear Algebra and Its Applications 5th Edition

Full solutions for Linear Algebra and Its Applications | 5th Edition

ISBN: 9780321982384

Solutions for Chapter 4.8

Solutions for Chapter 4.8
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ISBN: 9780321982384

This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since 37 problems in chapter 4.8 have been answered, more than 40379 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Chapter 4.8 includes 37 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• 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.

• Big formula for n by n determinants.

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

• 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.

• 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

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• Graph G.

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• 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.

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• 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).

• Rotation matrix

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

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