 9.8.1: If X(t) is any fundamental matrix for x = Ax, show that the transit...
 9.8.2: For 24, use the techniques from Section 9.4 and Section 9.5 to dete...
 9.8.3: For 24, use the techniques from Section 9.4 and Section 9.5 to dete...
 9.8.4: For 24, use the techniques from Section 9.4 and Section 9.5 to dete...
 9.8.5: For 57, find n linearly independent solutions to x = Ax of the form...
 9.8.6: For 57, find n linearly independent solutions to x = Ax of the form...
 9.8.7: For 57, find n linearly independent solutions to x = Ax of the form...
 9.8.8: For 810, solve x = Ax by determining n linearly independent solutio...
 9.8.9: For 810, solve x = Ax by determining n linearly independent solutio...
 9.8.10: For 810, solve x = Ax by determining n linearly independent solutio...
 9.8.11: The matrix A = 0 10 0 1 00 0 1 00 1 0 11 0 has characteristic polyn...
Solutions for Chapter 9.8: Matrix Exponential Function and Systems of Differential Equations
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 9.8: Matrix Exponential Function and Systems of Differential Equations
Get Full SolutionsSince 11 problems in chapter 9.8: Matrix Exponential Function and Systems of Differential Equations have been answered, more than 20153 students have viewed full stepbystep solutions from this chapter. Chapter 9.8: Matrix Exponential Function and Systems of Differential Equations includes 11 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. This textbook survival guide was created for the textbook: Differential Equations, edition: 4.

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

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.

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

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 addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).