 7.6.1: Let A = 1 1 1 1 (a) Apply one iteration of the power method to A, w...
 7.6.2: Let A = 2 1 0 1 3 1 0 1 2 and u0 = 1 1 1 (a) Apply the power method...
 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 = 5 2 2 2 1 2 3 4 2 (a) Verify that 1 = 4 is an eigenvalue of...
 7.6.6: Let A be an n n matrix with distinct real eigenvalues 1, 2, . . . ,...
 7.6.7: Let x = (x1, . . . , xn)T be an eigenvector of A belonging to . Sho...
 7.6.8: Let be an eigenvalue of an n n matrix A. Show that  aj j _n i=1 i...
 7.6.9: Let A be a matrix with eigenvalues 1, . . . , n and let be an eigen...
 7.6.10: Let Ak = Qk Rk , k = 1, 2, . . . be the sequence of matrices derive...
 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  8th Edition
ISBN: 9780136009290
Solutions for Chapter 7.6: The Eigenvalue Problem
Get Full SolutionsThis 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: 9780136009290. Since 13 problems in chapter 7.6: The Eigenvalue Problem have been answered, more than 5065 students have viewed full stepbystep solutions from this chapter. Chapter 7.6: The Eigenvalue Problem includes 13 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

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.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.