 7.8.1: In each of 1 through 4: (a) Draw a direction field and sketch a few...
 7.8.2: In each of 1 through 4: (a) Draw a direction field and sketch a few...
 7.8.3: In each of 1 through 4: (a) Draw a direction field and sketch a few...
 7.8.4: In each of 1 through 4: (a) Draw a direction field and sketch a few...
 7.8.5: In each of 5 and 6 find the general solution of the given system of...
 7.8.6: In each of 5 and 6 find the general solution of the given system of...
 7.8.7: In each of 7 through 10: (a) Find the solution of the given initial...
 7.8.8: In each of 7 through 10: (a) Find the solution of the given initial...
 7.8.9: In each of 7 through 10: (a) Find the solution of the given initial...
 7.8.10: In each of 7 through 10: (a) Find the solution of the given initial...
 7.8.11: In each of 11 and 12: (a) Find the solution of the given initial va...
 7.8.12: In each of 11 and 12: (a) Find the solution of the given initial va...
 7.8.13: In each of 13 and 14 solve the given system of equations by the met...
 7.8.14: In each of 13 and 14 solve the given system of equations by the met...
 7.8.15: Show that all solutions of the system x = a b c d x approach zero a...
 7.8.16: Consider again the electric circuit in of Section 7.6. This circuit...
 7.8.17: Consider the system x = Ax = 111 2 1 1 324 x.(a) Show that r = 2 is...
 7.8.18: Consider the system x = Ax = 5 3 2 8 5 4 433 x. (i) (a) Show that r...
 7.8.19: Let J = 1 0 , where is an arbitrary real number. (a) Find J2 , J3 ,...
 7.8.20: Let J = 0 0 0 1 0 0 , where is an arbitrary real number. (a) Find J...
 7.8.21: Let J = 1 0 0 1 0 0 , where is an arbitrary real number.(a) Find J2...
Solutions for Chapter 7.8: Repeated Eigenvalues
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 7.8: Repeated Eigenvalues
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Since 21 problems in chapter 7.8: Repeated Eigenvalues have been answered, more than 14308 students have viewed full stepbystep solutions from this chapter. Chapter 7.8: Repeated Eigenvalues includes 21 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

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

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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.

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