 5.1.1E: Is an eigenvalue of ?Why or why not?
 5.1.2E: Is an eigenvalue of ?Why or why not?
 5.1.3E: Is an eigenvector of ?If so, find the eigenvalue.
 5.1.4E: Is an eigenvector of ?so, find the eigenvalue.
 5.1.5E: If so, find the eigenvalue.
 5.1.6E: If so, find the eigenvalue.
 5.1.7E: If so, find one corresponding eigenvector.
 5.1.8E: If so, find one corresponding eigenvector.
 5.1.9E: In Exercises 9–16, find a basis for the eigenspace corresponding to...
 5.1.10E: In Exercises 9–16, find a basis for the eigenspace corresponding to...
 5.1.11E: In Exercises 9–16, find a basis for the eigenspace corresponding to...
 5.1.12E: In Exercises 9–16, find a basis for the eigenspace corresponding to...
 5.1.13E: In Exercises 9–16, find a basis for the eigenspace corresponding to...
 5.1.14E: In Exercises 9–16, find a basis for the eigenspace corresponding to...
 5.1.15E: In Exercises 9–16, find a basis for the eigenspace corresponding to...
 5.1.16E: In Exercises 9–16, find a basis for the eigenspace corresponding to...
 5.1.17E: Find the eigenvalues of the matrices in Exercises 17 and 18.
 5.1.18E: Find the eigenvalues of the matrices in Exercises 17 and 18.
 5.1.19E: find one eigenvalue, with no calculation. Justify your answer.
 5.1.20E: Without calculation, find one eigenvalue and two linearly independe...
 5.1.21E: In Exercises 21 and 22, A is an n × n matrix. Mark each statement T...
 5.1.22E: In Exercises 21 and 22, A is an n × n matrix. Mark each statement T...
 5.1.23E: Explain why a 2 × 2 matrix can have at most two distinct eigenvalue...
 5.1.24E: Construct an example of a 2 × 2 matrix with only one distinct eigen...
 5.1.25E: Let be an eigenvalue of an invertible matrix A. Show that is an eig...
 5.1.26E: Show that if A2 is the zero matrix, then the only eigenvalue of A i...
 5.1.27E: Show that is an eigenvalue of A if and only if is an eigenvalue of ...
 5.1.28E: Use Exercise 27 to complete the proof of Theorem 1 for the case in ...
 5.1.29E: Consider an n × n matrix A with the property that the row sums all ...
 5.1.30E: Consider an n × n matrix A with the property that the column sums a...
 5.1.31E: In Exercises 31 and 32, let A be the matrix of the linear transform...
 5.1.32E: In Exercises 31 and 32, let A be the matrix of the linear transform...
 5.1.33E: Let u and v be eigenvectors of a matrix A, with corresponding eigen...
 5.1.34E: Describe how you might try to build a solution of a difference equa...
 5.1.35E: Let u and v be the vectors shown in the figure, and suppose u and v...
 5.1.36E: Repeat Exercise 35, assuming u and v are eigenvectors of A that cor...
 5.1.37E: [M] In Exercises 37–40, use a matrix program to find the eigenvalue...
 5.1.38E: [M] In Exercises 37–40, use a matrix program to find the eigenvalue...
 5.1.39E: [M] In Exercises 37–40, use a matrix program to find the eigenvalue...
 5.1.40E: [M] In Exercises 37–40, use a matrix program to find the eigenvalue...
Solutions for Chapter 5.1: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 5.1
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Since 40 problems in chapter 5.1 have been answered, more than 80216 students have viewed full stepbystep solutions from this chapter. Chapter 5.1 includes 40 full stepbystep solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Iterative method.
A sequence of steps intended to approach the desired solution.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Outer product uv T
= column times row = rank one matrix.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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 ().

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.