 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 twoparameter family of linear systems dY dt = a 1 b 1...
 3.7.9: Consider the twoparameter family of linear systems dY dt = a b b a...
 3.7.10: Consider the twoparameter 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 TRACEDETERMINANT PLANE
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 3.7: THE TRACEDETERMINANT PLANE
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989. Since 14 problems in chapter 3.7: THE TRACEDETERMINANT PLANE have been answered, more than 15535 students have viewed full stepbystep solutions from this chapter. Chapter 3.7: THE TRACEDETERMINANT PLANE includes 14 full stepbystep 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.

Cholesky factorization
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.

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].

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 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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Similar matrices A and B.
Every B = MI 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.