 6.1: The top matrix on the menu is the diagonal matrix A = 5 4 0 0 3 4 I...
 6.2: The third matrix on the menu is just the identity matrix I. Howdo x...
 6.3: The fourth matrix has 0s on the diagonal and 1s in the offdiagonal...
 6.4: The next matrix in the eigshow menu looks the same as the previous ...
 6.5: Investigate the next three matrices on the menu (the sixth, seventh...
 6.6: Investigate the ninth matrix on the menu. What can you conclude abo...
 6.7: Investigate the next three matrices on the menu. You should note th...
 6.8: The last item on the eigshow menu will generate a random 2 2 matrix...
 6.9: Consider the application relating to critical loads for a beam from...
 6.10: Construct a symmetric matrix A by setting A = round(5 rand(6)); A =...
 6.11: Set A = ones(10) + eye(10) (a) What is the rank of A I ? Why must =...
 6.12: Consider the matrices A = 5 3 3 5 and B = 5 3 3 5 Note that the two...
 6.13: Set B = toeplitz(0: 1: 3, 0: 3) The matrix B is not symmetric and h...
 6.14: Set C = triu(ones(4), 1) + diag([1, 1],2) and [X, D] = eig(C) Compu...
 6.15: Construct a defective matrix by setting A = ones(6); A = Atril(A)tr...
 6.16: Generate a matrix A by setting B = [1, 1; 1, 1 ], A = [zeros(2), ey...
 6.17: Suppose that 10,000 men and 10,000 women settle on an island in the...
 6.18: Set S = round(10 rand(5)); S = triu(S, 1) + eye(5) S = S_ S T = inv...
 6.19: Construct a complex Hermitian matrix by setting j = sqrt(1); A = ra...
 6.20: Let A be a nonsingular 2 2 matrix with singular value decomposition...
 6.21: Set A = [1, 1; 0.5,0.5] and use MATLAB to verify each of statements...
 6.22: Use the following MATLAB commands to construct a symbolic function:...
 6.23: Set C = ones(6) + 7 eye(6) and [X, D] = eig(C) (a) Even though = 7 ...
 6.24: For various values of k, form a k k matrix A by setting D = diag(on...
 6.25: For any positive integer n, the MATLAB command P = pascal(n) will g...
Solutions for Chapter 6: Eigenvalues
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 6: Eigenvalues
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. Chapter 6: Eigenvalues includes 25 full stepbystep solutions. Since 25 problems in chapter 6: Eigenvalues have been answered, more than 7217 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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!

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

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.

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.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.