 9.5.1: For 114, determine the general solution to the system x = Ax for th...
 9.5.2: For 114, determine the general solution to the system x = Ax for th...
 9.5.3: For 114, determine the general solution to the system x = Ax for th...
 9.5.4: For 114, determine the general solution to the system x = Ax for th...
 9.5.5: For 114, determine the general solution to the system x = Ax for th...
 9.5.6: For 114, determine the general solution to the system x = Ax for th...
 9.5.7: For 114, determine the general solution to the system x = Ax for th...
 9.5.8: For 114, determine the general solution to the system x = Ax for th...
 9.5.9: For 114, determine the general solution to the system x = Ax for th...
 9.5.10: For 114, determine the general solution to the system x = Ax for th...
 9.5.11: For 114, determine the general solution to the system x = Ax for th...
 9.5.12: For 114, determine the general solution to the system x = Ax for th...
 9.5.13: For 114, determine the general solution to the system x = Ax for th...
 9.5.14: For 114, determine the general solution to the system x = Ax for th...
 9.5.15: For 1516, solve the initialvalue problem.
 9.5.16: For 1516, solve the initialvalue problem.
 9.5.17: Show that if the vector differential equation x = Ax has a solution...
 9.5.18: Let A be a 2 2 real matrix. Prove that all solutions to x = Ax sati...
 9.5.19: Let A be a 2 2 real matrix. Prove that all solutions to x = Ax sati...
 9.5.20: This problem outlines a proof of Theorem 9.5.4 using results from t...
Solutions for Chapter 9.5: Vector Differential Equations: Defective Coefficient Matrix
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 9.5: Vector Differential Equations: Defective Coefficient Matrix
Get Full SolutionsSince 20 problems in chapter 9.5: Vector Differential Equations: Defective Coefficient Matrix have been answered, more than 19770 students have viewed full stepbystep solutions from this chapter. Differential Equations was written by and is associated to the ISBN: 9780321964670. Chapter 9.5: Vector Differential Equations: Defective Coefficient Matrix includes 20 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations, edition: 4.

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).

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

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.

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.)

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].

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.