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Solutions for Chapter 6.1: Introduction to Eigenvalues

Full solutions for Introduction to Linear Algebra | 4th Edition

ISBN: 9780980232714

Solutions for Chapter 6.1: Introduction to Eigenvalues

Solutions for Chapter 6.1
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Textbook: Introduction to Linear Algebra
Edition: 4
Author: Gilbert Strang
ISBN: 9780980232714

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.1: Introduction to Eigenvalues includes 37 full step-by-step solutions. Since 37 problems in chapter 6.1: Introduction to Eigenvalues have been answered, more than 10884 students have viewed full step-by-step solutions from this chapter. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • 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.

  • 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.

  • Indefinite matrix.

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

  • Jordan form 1 = M- 1 AM.

    If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • 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).

  • Multiplication Ax

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

  • Multiplier eij.

    The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

  • Normal matrix.

    If N NT = NT N, then N has orthonormal (complex) eigenvectors.

  • Particular solution x p.

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

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def 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!

  • Skew-symmetric matrix K.

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

  • 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}.

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.

  • Wavelets Wjk(t).

    Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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