- 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. How do ...
- 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: Consider the matrices A = 5 3 3 5 and B = 5 3 3 5 Note that the two...
- 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 an...
- 6.14: Set C = triu(ones(4), 1) + diag([1, 1], 2) and [X, D] = eig(C) Comp...
- 6.15: Set C = triu(ones(4), 1) + diag([1, 1], 2) and [X, D] = eig(C) Comp...
- 6.16: Generate a matrix A by setting B = [ 1, 1; 1, 1 ], A = [zeros(2), e...
- 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: Set S = round(10 rand(5)); S = triu(S, 1) + eye(5) S = S S T = inv(...
- 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 statement...
- 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 an k k matrix A by setting D = diag(o...
- 6.25: For various values of k, form an k k matrix A by setting D = diag(o...
Solutions for Chapter 6: Eigenvalues
Full solutions for Linear Algebra with Applications | 9th Edition
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.
Remove row i and column j; multiply the determinant by (-I)i + j •
Column space C (A) =
space of all combinations of the columns of A.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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.
A sequence of steps intended to approach the desired solution.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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 ().
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Solvable system Ax = b.
The right side b is in the column space of A.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.