×
Log in to StudySoup
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide

Solutions for Chapter 3.3: Nonhomogeneous Linear ODEs

Advanced Engineering Mathematics | 9th Edition | ISBN: 9780471488859 | Authors: Erwin Kreyszig

Full solutions for Advanced Engineering Mathematics | 9th Edition

ISBN: 9780471488859

Advanced Engineering Mathematics | 9th Edition | ISBN: 9780471488859 | Authors: Erwin Kreyszig

Solutions for Chapter 3.3: Nonhomogeneous Linear ODEs

This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Since 20 problems in chapter 3.3: Nonhomogeneous Linear ODEs have been answered, more than 46347 students have viewed full step-by-step solutions from this chapter. Chapter 3.3: Nonhomogeneous Linear ODEs includes 20 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • Diagonal matrix D.

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

  • Diagonalizable matrix A.

    Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

  • 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.

  • 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].

  • Free variable Xi.

    Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • 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.

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • 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.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • Pseudoinverse A+ (Moore-Penrose inverse).

    The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Spectral Theorem A = QAQT.

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

  • Toeplitz matrix.

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

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

×
Log in to StudySoup
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Calculus and Pre Calculus - Textbook Survival Guide
×
Reset your password