 7.1.1: In Exercises I through 4, let A be an invertible n x n matrix and v...
 7.1.2: In Exercises I through 4, let A be an invertible n x n matrix and v...
 7.1.3: In Exercises I through 4, let A be an invertible n x n matrix and v...
 7.1.4: In Exercises I through 4, let A be an invertible n x n matrix and v...
 7.1.5: If a vector v is an eigenvector of both A and B, is 5 necessarily a...
 7.1.6: If a vector v is an eigenvector of both A and By is 5 necessarily a...
 7.1.7: If v is an eigenvector of the nxn matrix A with associated eigenval...
 7.1.8: Find all 2 x 2 matrices for which e\ = vector with associated eigen...
 7.1.9: Find all 2 x 2 matrices for which e\ is an eigenvector.
 7.1.10: Find all 2 x 2 matrices for which with associated eigenvalue 5.
 7.1.11: Find all 2 x 2 matrices for which with associated eigenvalue 1.
 7.1.12: Consider the matrix A =3 4. Show that 2 and 4are eigenvalues of A a...
 7.1.13: Show that 4 is an eigenvalue of A = andfind all corresponding eigen...
 7.1.14: Find all 4 x 4 matrices for which ei is an eigenvector.
 7.1.15: Arguing geometrically; find all eigenvectors and eigenvalues of the...
 7.1.16: Arguing geometrically; find all eigenvectors and eigenvalues of the...
 7.1.17: Arguing geometrically; find all eigenvectors and eigenvalues of the...
 7.1.18: Arguing geometrically; find all eigenvectors and eigenvalues of the...
 7.1.19: Arguing geometrically; find all eigenvectors and eigenvalues of the...
 7.1.20: Arguing geometrically; find all eigenvectors and eigenvalues of the...
 7.1.21: Arguing geometrically; find all eigenvectors and eigenvalues of the...
 7.1.22: Arguing geometrically; find all eigenvectors and eigenvalues of the...
 7.1.23: a. Consider an invertible matrix S = [5 v2 ... 5n]. Find Hint: Wha...
 7.1.24: In Exercises 24 through 29, consider a dynamical systemx(t + 1) = A...
 7.1.25: In Exercises 24 through 29, consider a dynamical systemx(t + 1) = A...
 7.1.26: In Exercises 24 through 29, consider a dynamical systemx(t + 1) = A...
 7.1.27: In Exercises 24 through 29, consider a dynamical systemx(t + 1) = A...
 7.1.28: In Exercises 24 through 29, consider a dynamical systemx(t + 1) = A...
 7.1.29: In Exercises 24 through 29, consider a dynamical systemx(t + 1) = A...
 7.1.30: In Exercises 30 through 32, consider the dynamical system1.1 0x(t +...
 7.1.31: In Exercises 30 through 32, consider the dynamical system1.1 0x(t +...
 7.1.32: In Exercises 30 through 32, consider the dynamical system1.1 0x(t +...
 7.1.33: Find a 2 x 2 matrix A such that* 2r 6f X = [2' + 6'Jis a trajectory...
 7.1.34: Suppose v is an eigenvector of the n x n matrix A, with eigenvalue ...
 7.1.35: Show that similar matrices have the same eigenvalues. (Hint: If v i...
 7.1.36: Find a 2 x 2 matrix A such that are eigen vectors of A, with eigen...
 7.1.37: Consider the matrix3 4a. Use the geometric interpretation of this t...
 7.1.38: We are told that4 1 1 5 0 3 1 1 21 1; what is the associated ...
 7.1.39: Find a basis of the linear space V of all 2 x 2 matrices A for whic...
 7.1.40: Find a basis of the linear space V of all 2 x 2 matrices A ll for w...
 7.1.41: Find a basis of the linear space V of all 2 x 2 matrices Afor which...
 7.1.42: Consider the linear space V of all n x n matrices for which all the...
 7.1.43: Consider the linear space V of all n x n matrices for which all the...
 7.1.44: For m < n, find the dimension of the space of all n x n matrices A ...
 7.1.45: If v is any nonzero vector in IR2, what is the dimension of the spa...
 7.1.46: In all parts of this problem, let V be the linear space of all 2 x ...
 7.1.47: Consider an nxn matrix A. A subspace V of Rn is said to be Ainvari...
 7.1.48: a. Give an example of a 3 x 3 matrix A with as many nonzero entries...
 7.1.49: Consider the coyotesroadrunner system discussed in this section. F...
 7.1.50: Two interacting populations of hares and foxes can be modeled by th...
 7.1.51: Two interacting populations of coyots and roadrunners can be modele...
 7.1.52: Imagine that you are diabetic and have to pay close attention to ho...
 7.1.53: Three holy men (lets call them Abraham, Benjamin, and Chaim) put li...
 7.1.54: Consider the growth of a lilac bush. The state of this lilac bush f...
Solutions for Chapter 7.1: Dynamical Systems and Eigenvectors: An Introductory Example
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 7.1: Dynamical Systems and Eigenvectors: An Introductory Example
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. Since 54 problems in chapter 7.1: Dynamical Systems and Eigenvectors: An Introductory Example have been answered, more than 15453 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Chapter 7.1: Dynamical Systems and Eigenvectors: An Introductory Example includes 54 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

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).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.