 3.4.1: In Exercises 18 determine the characteristic polynomials, eigenval...
 3.4.2: In Exercises 18 determine the characteristic polynomials, eigenval...
 3.4.3: In Exercises 18 determine the characteristic polynomials, eigenval...
 3.4.4: In Exercises 18 determine the characteristic polynomials, eigenval...
 3.4.5: In Exercises 18 determine the characteristic polynomials, eigenval...
 3.4.6: In Exercises 18 determine the characteristic polynomials, eigenval...
 3.4.7: In Exercises 18 determine the characteristic polynomials, eigenval...
 3.4.8: In Exercises 18 determine the characteristic polynomials, eigenval...
 3.4.9: In Exercises 914 determine the characteristic polynomials, eigenva...
 3.4.10: In Exercises 914 determine the characteristic polynomials, eigenva...
 3.4.11: In Exercises 914 determine the characteristic polynomials, eigenva...
 3.4.12: In Exercises 914 determine the characteristic polynomials, eigenva...
 3.4.13: In Exercises 914 determine the characteristic polynomials, eigenva...
 3.4.14: In Exercises 914 determine the characteristic polynomials, eigenva...
 3.4.15: In Exercises 15 and 16 determine the characteristic polynomials, ei...
 3.4.16: In Exercises 15 and 16 determine the characteristic polynomials, ei...
 3.4.17: In Exercises 1719 determine the characteristic polynomials, eigenv...
 3.4.18: In Exercises 1719 determine the characteristic polynomials, eigenv...
 3.4.19: In Exercises 1719 determine the characteristic polynomials, eigenv...
 3.4.20: Show that the following matrix has no real eigenvalues and thus no ...
 3.4.21: Show that the following matrix has no real eigenvalues. Interpret y...
 3.4.22: Find the eigenvalues and eigenvectors of the identity matrix In In...
 3.4.23: LetA be then X n matrix having every element 1. Find the eigenvalue...
 3.4.24: Prove that if A is a diagonal matrix then its eigenvalues are the d...
 3.4.25: Prove that if A is an upper triangular matrix then its eigenvalues ...
 3.4.26: LetA be a square matrix. Prove thatA and A1 have the same eigenvalues.
 3.4.27: Prove that,\ = 0 is an eigenvalue of a matrix A if and only if A is...
 3.4.28: Prove that if the eigenvalues of a matrix A are A1, , Ano with corr...
 3.4.29: LetA be an invertible matrix with eigenvalue,\ having corresponding...
 3.4.30: Let A be a matrix with eigenvalue ,\ having corresponding eigenvect...
 3.4.31: A matrix A is said to be nilpotent if A k = 0 for some integer k. P...
 3.4.32: Prove that the constant term of the characteristic polynomial of a ...
 3.4.33: Determine the eigenvalues and corresponding eigenvectors of the mat...
 3.4.34: There is a theorem called The CayleyHamilton Theorem, which states...
 3.4.35: State (with a brief explanation) whether the following statements a...
Solutions for Chapter 3.4: Eigenvalues and Eigenvectors
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 3.4: Eigenvalues and Eigenvectors
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.4: Eigenvalues and Eigenvectors includes 35 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. Since 35 problems in chapter 3.4: Eigenvalues and Eigenvectors have been answered, more than 25776 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Column space C (A) =
space of all combinations of the columns of A.

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

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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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