 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 o...
 7.8.6: In each of 5 and 6, find the general solution of the given system o...
 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 val...
 7.8.12: In each of 11 and 12:(a) Find the solution of the given initial val...
 7.8.13: In each of 13 and 14, solve the given system of equations by the me...
 7.8.14: In each of 13 and 14, solve the given system of equations by the me...
 7.8.15: Show that all solutions of the systemx=a bc dxapproach zero as t if...
 7.8.16: Consider again the electric circuit in of Section 7.6. This circuit...
 7.8.17: . Consider again the systemx= Ax =1 11 3x (i)that we discussed in E...
 7.8.18: Consider the systemx= Ax =1112 1 1324 x. (i)(a) Show that r = 2 is ...
 7.8.19: Consider the systemx= Ax =5 3 28 5 4433 x. (i)(a) Show that r = 1 i...
 7.8.20: Let J = 10 , where is an arbitrary real number.(a) Find J2, J3, and...
 7.8.21: LetJ = 0 00 10 0 ,where is an arbitrary real number.(a) Find J2, J3...
 7.8.22: LetJ = 1 00 10 0 ,where is an arbitrary real number.(a) Find J2, J3...
Solutions for Chapter 7.8: Repeated Eigenvalues
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 7.8: Repeated Eigenvalues
Get Full SolutionsElementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. Since 22 problems in chapter 7.8: Repeated Eigenvalues have been answered, more than 18269 students have viewed full stepbystep solutions from this chapter. Chapter 7.8: Repeated Eigenvalues includes 22 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

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.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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.

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

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

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).