 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 8097 students have viewed full stepbystep solutions from this chapter.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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 •

Outer product uv T
= column times row = rank one matrix.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.