- 3.7.1: Construct a table of the possible linear systems as follows: (a) Th...
- 3.7.2: dY dt = a 1 2 0 Y 3
- 3.7.3: dY dt = a a2 + a 1 a Y 4
- 3.7.4: dY dt = a a 1 0 Y 5.
- 3.7.5: dY dt = a 1 a2 1 0 Y
- 3.7.6: dY dt = 2 0 a 3 Y 7.
- 3.7.7: dY dt = a 1 a a Y 8
- 3.7.8: Consider the two-parameter family of linear systems dY dt = a 1 b 1...
- 3.7.9: Consider the two-parameter family of linear systems dY dt = a b b a...
- 3.7.10: Consider the two-parameter family of linear systems dY dt = a b b a...
- 3.7.11: Consider d2 y dt2 + b dy dt + 3y = 0. That is, fix m = 1 and k = 3,...
- 3.7.12: Consider d2 y dt2 + 2 dy dt + ky = 0. That is, fix m = 1 and b = 2,...
- 3.7.13: Consider m d2 y dt2 + dy dt + 2y = 0. That is, fix b = 1 and k = 2,...
- 3.7.14: Using the DETools program TDPlaneQuiz, describe the path through th...
Solutions for Chapter 3.7: THE TRACE-DETERMINANT PLANE
Full solutions for Differential Equations 00 | 4th Edition
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.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
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].
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.
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.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Solvable system Ax = b.
The right side b is in the column space of A.
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}.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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.