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Solutions for Chapter 3.7: THE TRACE-DETERMINANT PLANE

Differential Equations 00 | 4th Edition | ISBN: 9780495561989 | Authors: Paul (Paul Blanchard) Blanchard, Robert L. Devaney, Glen R. Hall

Full solutions for Differential Equations 00 | 4th Edition

ISBN: 9780495561989

Differential Equations 00 | 4th Edition | ISBN: 9780495561989 | Authors: Paul (Paul Blanchard) Blanchard, Robert L. Devaney, Glen R. Hall

Solutions for Chapter 3.7: THE TRACE-DETERMINANT PLANE

This 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 TRACE-DETERMINANT PLANE have been answered, more than 15535 students have viewed full step-by-step solutions from this chapter. Chapter 3.7: THE TRACE-DETERMINANT PLANE includes 14 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • 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 = 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.

  • Schwarz inequality

    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.

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