- 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
Solutions for Chapter 9.8: Matrix Exponential Function and Systems of Differential EquationsGet Full Solutions
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 IIA-1 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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + 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.
Invert A by row operations on [A I] to reach [I A-I].
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 A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
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)j-I and det V = product of (Xk - Xi) for k > i.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).