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# Solutions for Chapter 7.1: Eigenvalues and Eigenvectors

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

ISBN: 9780471669593

Solutions for Chapter 7.1: Eigenvalues and Eigenvectors

Solutions for Chapter 7.1
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##### ISBN: 9780471669593

This expansive textbook survival guide covers the following chapters and their solutions. Since 40 problems in chapter 7.1: Eigenvalues and Eigenvectors have been answered, more than 10232 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. Chapter 7.1: Eigenvalues and Eigenvectors includes 40 full step-by-step solutions.

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

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• 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.

• 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.

• Fast Fourier Transform (FFT).

A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.

• 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.

• Indefinite matrix.

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

• 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.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• 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.

• Outer product uv T

= column times row = rank one matrix.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• 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.

• Standard basis for Rn.

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