- 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 off-diagonal...
- 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
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!
A = CTC = (L.J]))(L.J]))T for positive definite A.
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].
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!).
Invert A by row operations on [A I] to reach [I A-I].
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, off-diagonal entries = 0.
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.
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.
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.
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 ().
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I 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.