 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 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: 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
ISBN: 9780321962218
Solutions for Chapter 6: Eigenvalues
Get Full SolutionsSince 25 problems in chapter 6: Eigenvalues have been answered, more than 11962 students have viewed full stepbystep solutions from this chapter. Chapter 6: Eigenvalues includes 25 full stepbystep solutions. 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: 9780321962218. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9.

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.

Cofactor Cij.
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.

Fundamental Theorem.
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 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.

Iterative method.
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.

Network.
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.

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).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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 ().

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.