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Solutions for Chapter 7.2: Diagonalization and Similar Matrices

Full solutions for Elementary Linear Algebra with Applications | 9th Edition

ISBN: 9780471669593

Solutions for Chapter 7.2: Diagonalization and Similar Matrices

Solutions for Chapter 7.2
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Textbook: Elementary Linear Algebra with Applications
Edition: 9
Author: Howard Anton, Chris Rorres
ISBN: 9780471669593

Chapter 7.2: Diagonalization and Similar Matrices includes 40 full step-by-step solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. Since 40 problems in chapter 7.2: Diagonalization and Similar Matrices have been answered, more than 9189 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Big formula for n by n determinants.

    Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Fourier matrix F.

    Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

  • 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.

  • Least squares solution X.

    The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

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

    Vector addition and scalar multiplication.

  • 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.

  • 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.

  • 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 •

  • Positive definite matrix A.

    Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as 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!

  • Standard basis for Rn.

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

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B II·

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

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