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Solutions for Chapter 6.5: Positive Definite Matrices

Full solutions for Introduction to Linear Algebra | 4th Edition

ISBN: 9780980232714

Solutions for Chapter 6.5: Positive Definite Matrices

Solutions for Chapter 6.5
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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. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. Chapter 6.5: Positive Definite Matrices includes 35 full step-by-step solutions. Since 35 problems in chapter 6.5: Positive Definite Matrices have been answered, more than 8145 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4.

Key Math Terms and definitions covered in this textbook
  • Adjacency matrix of a graph.

    Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • Cramer's Rule for Ax = b.

    B j has b replacing column j of A; x j = det B j I det A

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • Ellipse (or ellipsoid) x T Ax = 1.

    A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • 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.

  • 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.

  • 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 .

  • Network.

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

  • Norm

    IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

  • Normal equation AT Ax = ATb.

    Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

  • Orthonormal vectors q 1 , ... , q n·

    Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • 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.

  • Sum V + W of subs paces.

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

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

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