 7.6.1: For 15, determine how many Jordan canonical forms are with the give...
 7.6.2: For 15, determine how many Jordan canonical forms are with the give...
 7.6.3: For 15, determine how many Jordan canonical forms are with the give...
 7.6.4: For 15, determine how many Jordan canonical forms are with the give...
 7.6.5: For 15, determine how many Jordan canonical forms are with the give...
 7.6.6: For 67, determine how many Jordan canonical forms are possible with...
 7.6.7: For 67, determine how many Jordan canonical forms are possible with...
 7.6.8: If it is known that (A 5I) 2 = 0 for the matrix in 7, how many Jord...
 7.6.9: Let A be a 5 5 matrix with eigenvalues 1,1, 1,2,2, where 1 = 2. (a)...
 7.6.10: Suppose A is a 6 6 matrix with eigenvalue (of multiplicity 6). If i...
 7.6.11: For 1114, the characteristic polynomial p() for a square matrix A i...
 7.6.12: For 1114, the characteristic polynomial p() for a square matrix A i...
 7.6.13: For 1114, the characteristic polynomial p() for a square matrix A i...
 7.6.14: For 1114, the characteristic polynomial p() for a square matrix A i...
 7.6.15: Which of the matrices in the set S in have a set of exaxctly five (...
 7.6.16: Give an example of a 22 matrix A that has a generalized eigenvector...
 7.6.17: Give an example of a 33 matrix A that has a generalized eigenvector...
 7.6.18: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.19: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.20: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.21: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.22: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.23: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.24: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.25: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.26: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.27: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.28: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.29: For 1829, find the Jordan canonical form J for the matrix A, and de...
 7.6.30: For 3032, find the Jordan canonical form J for the matrix A. You ne...
 7.6.31: For 3032, find the Jordan canonical form J for the matrix A. You ne...
 7.6.32: For 3032, find the Jordan canonical form J for the matrix A. You ne...
 7.6.33: For 3335, use Jordan canonical forms to determine whether the given...
 7.6.34: For 3335, use Jordan canonical forms to determine whether the given...
 7.6.35: For 3335, use Jordan canonical forms to determine whether the given...
 7.6.36: For 3641, determine the general solution to the system x = Ax for t...
 7.6.37: For 3641, determine the general solution to the system x = Ax for t...
 7.6.38: For 3641, determine the general solution to the system x = Ax for t...
 7.6.39: For 3641, determine the general solution to the system x = Ax for t...
 7.6.40: For 3641, determine the general solution to the system x = Ax for t...
 7.6.41: For 3641, determine the general solution to the system x = Ax for t...
 7.6.42: Solve the initialvalue problem x = Ax, where A = 2 1 1 4 , x(0) = ...
 7.6.43: Prove that if A are B are n n matrices with the same Jordan canonic...
 7.6.44: Let A be a square matrix with characteristic polynomial p() = 3. Us...
 7.6.45: (a) Let J be a Jordan block. Prove that the Jordan canonical form o...
Solutions for Chapter 7.6: Jordan Canonical Forms
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 7.6: Jordan Canonical Forms
Get Full SolutionsSince 45 problems in chapter 7.6: Jordan Canonical Forms have been answered, more than 21320 students have viewed full stepbystep solutions from this chapter. Chapter 7.6: Jordan Canonical Forms includes 45 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations, edition: 4. Differential Equations was written by and is associated to the ISBN: 9780321964670. This expansive textbook survival guide covers the following chapters and their solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

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 •

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

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.

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

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.