 7.6.1: Let A = 1 1 1 1 (a) Apply one iteration of the power method to A wi...
 7.6.2: . Let A = 210 131 012 and u0 = 1 1 1 (a) Apply the power method to ...
 7.6.3: Let A = 1 2 1 1 and u0 = 1 1 (a) Compute u1, u2, u3, and u4, using ...
 7.6.4: Let A = A1 = 1 1 1 3 Compute A2 and A3, using the QR algorithm. Com...
 7.6.5: Let A = 522 2 1 2 3 4 2 (a) Verify that 1 = 4 is an eigenvalue of A...
 7.6.6: Let A be an n n matrix with distinct real eigenvalues 1, 2, ... , n...
 7.6.7: Let x = (x1, ... , xn) T be an eigenvector of A belonging to . Show...
 7.6.8: Let be an eigenvalue of an n n matrix A. Show that for some index j...
 7.6.9: Let be an eigenvalue of an n n matrix A. Show that for some index j...
 7.6.10: Let Ak = QkRk, k = 1, 2, ... be the sequence of matrices derived fr...
 7.6.11: Let Pk and Uk be defined as in Exercise 10. Show that (a) Pk+1Uk+1 ...
 7.6.12: Let Rk be a k k upper triangular matrix and suppose that RkUk = UkD...
 7.6.13: Let R be an n n upper triangular matrix whose diagonal entries are ...
Solutions for Chapter 7.6: The Eigenvalue Problem
Full solutions for Linear Algebra with Applications  9th Edition
ISBN: 9780321962218
Solutions for Chapter 7.6: The Eigenvalue Problem
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. Since 13 problems in chapter 7.6: The Eigenvalue Problem have been answered, more than 10418 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218. Chapter 7.6: The Eigenvalue Problem includes 13 full stepbystep solutions.

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

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.

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 nl  An).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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.

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).

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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.