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Solutions for Chapter 11.4: Autonomous Systems as Mathematical Models

Advanced Engineering Mathematics | 6th Edition | ISBN: 9781284105902 | Authors: Dennis G. Zill

Full solutions for Advanced Engineering Mathematics | 6th Edition

ISBN: 9781284105902

Advanced Engineering Mathematics | 6th Edition | ISBN: 9781284105902 | Authors: Dennis G. Zill

Solutions for Chapter 11.4: Autonomous Systems as Mathematical Models

Solutions for Chapter 11.4
4 5 0 320 Reviews
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Textbook: Advanced Engineering Mathematics
Edition: 6
Author: Dennis G. Zill
ISBN: 9781284105902

Chapter 11.4: Autonomous Systems as Mathematical Models includes 22 full step-by-step solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Since 22 problems in chapter 11.4: Autonomous Systems as Mathematical Models have been answered, more than 39021 students have viewed full step-by-step solutions from this chapter.

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.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Cholesky factorization

    A = CTC = (L.J]))(L.J]))T for positive definite A.

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • 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.

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • 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.

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • Lucas numbers

    Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

  • Norm

    IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

  • Nullspace N (A)

    = All solutions to Ax = O. Dimension n - r = (# columns) - rank.

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • Symmetric matrix A.

    The transpose is AT = A, and aU = a ji. A-I is also symmetric.

  • Toeplitz matrix.

    Constant down each diagonal = time-invariant (shift-invariant) filter.

  • 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.

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