 9.1.1: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.2: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.3: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.4: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.5: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.6: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.7: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.8: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.9: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.10: For each of the systems in 1 through 10: a. Find the eigenvalues an...
 9.1.11: In each of 11 through 13, determine the critical point x = x0, and ...
 9.1.12: In each of 11 through 13, determine the critical point x = x0, and ...
 9.1.13: In each of 11 through 13, determine the critical point x = x0, and ...
 9.1.14: The equation of motion of a springmass system with damping (see S...
 9.1.15: Consider the system x_ = Ax, and suppose that A has one zero eigenv...
 9.1.16: In this problem we indicate how to show that the trajectories are e...
 9.1.17: Consider the linear system dx dt = a11x + a12 y, dy dt = a21x + a22...
 9.1.18: Continuing 17, show that the critical point (0, 0) is a. Asymptotic...
 9.1.19: In this problem we illustrate how a 2 2 system with eigenvalues i c...
Solutions for Chapter 9.1: The Phase Plane: Linear Systems
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 9.1: The Phase Plane: Linear Systems
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Since 19 problems in chapter 9.1: The Phase Plane: Linear Systems have been answered, more than 12646 students have viewed full stepbystep solutions from this chapter. Chapter 9.1: The Phase Plane: Linear Systems includes 19 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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.