 5.6.1: Apply the method of undetermined coefficients to find a particular ...
 5.6.2: Apply the method of undetermined coefficients to find a particular ...
 5.6.3: Apply the method of undetermined coefficients to find a particular ...
 5.6.4: Apply the method of undetermined coefficients to find a particular ...
 5.6.5: Apply the method of undetermined coefficients to find a particular ...
 5.6.6: Apply the method of undetermined coefficients to find a particular ...
 5.6.7: Apply the method of undetermined coefficients to find a particular ...
 5.6.8: Apply the method of undetermined coefficients to find a particular ...
 5.6.9: Apply the method of undetermined coefficients to find a particular ...
 5.6.10: Apply the method of undetermined coefficients to find a particular ...
 5.6.11: Apply the method of undetermined coefficients to find a particular ...
 5.6.12: Apply the method of undetermined coefficients to find a particular ...
 5.6.13: Apply the method of undetermined coefficients to find a particular ...
 5.6.14: Apply the method of undetermined coefficients to find a particular ...
 5.6.15: 15 and 16 are similar to Example 2, but with two brine tanks (havin...
 5.6.16: 15 and 16 are similar to Example 2, but with two brine tanks (havin...
 5.6.17: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.18: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.19: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.20: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.21: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.22: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.23: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.24: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.25: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.26: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.27: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.28: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.29: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.30: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.31: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.32: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.33: In 17 through 34, use the method of variation of parameters (and pe...
 5.6.34: In 17 through 34, use the method of variation of parameters (and pe...
Solutions for Chapter 5.6: Matrix Exponentials and Linear Systems
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 5.6: Matrix Exponentials and Linear Systems
Get Full SolutionsSince 34 problems in chapter 5.6: Matrix Exponentials and Linear Systems have been answered, more than 15760 students have viewed full stepbystep solutions from this chapter. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.6: Matrix Exponentials and Linear Systems includes 34 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Solvable system Ax = b.
The right side b is in the column space of A.