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Complex Variables and Applications 9th Edition - Solutions by Chapter

Complex Variables and Applications | 9th Edition | ISBN: 9780073383170 | Authors: James Ward Brown

Full solutions for Complex Variables and Applications | 9th Edition

ISBN: 9780073383170

Complex Variables and Applications | 9th Edition | ISBN: 9780073383170 | Authors: James Ward Brown

Complex Variables and Applications | 9th Edition - Solutions by Chapter

This expansive textbook survival guide covers the following chapters: 12. This textbook survival guide was created for the textbook: Complex Variables and Applications, edition: 9. Since problems from 12 chapters in Complex Variables and Applications have been answered, more than 1989 students have viewed full step-by-step answer. Complex Variables and Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780073383170. The full step-by-step solution to problem in Complex Variables and Applications were answered by Sieva Kozinsky, our top Math solution expert on 12/23/17, 04:39PM.

Key Math Terms and definitions covered in this textbook
  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

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

    The n roots are the eigenvalues of A.

  • Cofactor Cij.

    Remove row i and column j; multiply the determinant by (-I)i + j •

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • 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.

  • Least squares solution X.

    The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Minimal polynomial of A.

    The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • Orthogonal matrix Q.

    Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

  • Pivot columns of A.

    Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

  • Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

    Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • Spectral Theorem A = QAQT.

    Real symmetric A has real A'S and orthonormal q's.

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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