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Solutions for Chapter 5.5: SturmLiouville Eigenvalue Problems

Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition | ISBN: 9780321797056 | Authors: Richard Haberman

Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition

ISBN: 9780321797056

Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition | ISBN: 9780321797056 | Authors: Richard Haberman

Solutions for Chapter 5.5: SturmLiouville Eigenvalue Problems

Solutions for Chapter 5.5
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Textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems
Edition: 5
Author: Richard Haberman
ISBN: 9780321797056

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.5: SturmLiouville Eigenvalue Problems includes 24 full step-by-step solutions. Since 24 problems in chapter 5.5: SturmLiouville Eigenvalue Problems have been answered, more than 8049 students have viewed full step-by-step solutions from this chapter. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056. This textbook survival guide was created for the textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, edition: 5.

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.

  • Complex conjugate

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

  • Fundamental Theorem.

    The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

  • Independent vectors VI, .. " vk.

    No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

  • Length II x II.

    Square root of x T x (Pythagoras in n dimensions).

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • Network.

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

  • Particular solution x p.

    Any solution to Ax = b; often x p has free variables = o.

  • 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.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

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

    P = aaT laTa has rank l.

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • 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!

  • Skew-symmetric matrix K.

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

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

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

    Spectral radius = max of IAi I.

  • Subspace S of V.

    Any vector space inside V, including V and Z = {zero vector only}.

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