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Solutions for Chapter 8.1: Symmetric Matrices

Full solutions for Linear Algebra with Applications | 4th Edition

ISBN: 9780136009269

Solutions for Chapter 8.1: Symmetric Matrices

Solutions for Chapter 8.1
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Textbook: Linear Algebra with Applications
Edition: 4
Author: Otto Bretscher
ISBN: 9780136009269

This expansive textbook survival guide covers the following chapters and their solutions. Since 50 problems in chapter 8.1: Symmetric Matrices have been answered, more than 14228 students have viewed full step-by-step solutions from this chapter. Chapter 8.1: Symmetric Matrices includes 50 full step-by-step solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4.

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

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • 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).

  • 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.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Hessenberg matrix H.

    Triangular matrix with one extra nonzero adjacent diagonal.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • 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).

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • Pivot.

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

  • 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 ().

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Schwarz inequality

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

  • Semidefinite matrix A.

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

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • Toeplitz matrix.

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

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