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Solutions for Chapter 3.5: SPECIAL CASES: REPEATED AND ZERO EIGENVALUES

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 3.5: SPECIAL CASES: REPEATED AND ZERO EIGENVALUES

Solutions for Chapter 3.5
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Textbook: Differential Equations 00
Edition: 4
Author: Paul (Paul Blanchard) Blanchard, Robert L. Devaney, Glen R. Hall
ISBN: 9780495561989

Differential Equations 00 was written by and is associated to the ISBN: 9780495561989. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.5: SPECIAL CASES: REPEATED AND ZERO EIGENVALUES includes 24 full step-by-step solutions. Since 24 problems in chapter 3.5: SPECIAL CASES: REPEATED AND ZERO EIGENVALUES have been answered, more than 16027 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

    Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Eigenvalue A and eigenvector x.

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

  • Gauss-Jordan method.

    Invert A by row operations on [A I] to reach [I A-I].

  • Hilbert matrix hilb(n).

    Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Left nullspace N (AT).

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

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • Nullspace N (A)

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

  • 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.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Reduced row echelon form R = rref(A).

    Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

  • Saddle point of I(x}, ... ,xn ).

    A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

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

    Spectral radius = max of IAi I.

  • Toeplitz matrix.

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

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