 3.1 1.1: Compute eAt for A equal
 3.1 1.2: Compute eAt for A equal
 3.1 1.3: Compute eAt for A equal
 3.1 1.4: Compute eAt for A equal
 3.1 1.5: Compute eAt for A equal
 3.1 1.6: Compute eAt for A equal
 3.1 1.7: Find A if
 3.1 1.8: In each of 81 1, determine whether the given matrix is a fundament...
 3.1 1.9: In each of 81 1, determine whether the given matrix is a fundament...
 3.1 1.10: In each of 81 1, determine whether the given matrix is a fundament...
 3.1 1.11: In each of 81 1, determine whether the given matrix is a fundament...
 3.1 1.12: Let +'(t) be the solution of the initialvalue problem x= Ax, x(0) ...
 3.1 1.13: Suppose that Y(t)=X(t)C, where X(t) and Y(t) are fundamental matrix...
 3.1 1.14: Let X(t) be a fundamental matrix solution of (I), and C a constant ...
 3.1 1.15: Let X(t) be a fundamental matrix solution of f =Ax. Prove that the ...
 3.1 1.16: Let X(t) be a fundamental matrix solution of x =Ax. Prove that X(t)...
 3.1 1.17: Here is an elegant proof of the identity eA'+B'= eA'eB' if ABBA. (...
Solutions for Chapter 3.1 1: Fundamental matrix solutions; e *'
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 3.1 1: Fundamental matrix solutions; e *'
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.1 1: Fundamental matrix solutions; e *' includes 17 full stepbystep solutions. Since 17 problems in chapter 3.1 1: Fundamental matrix solutions; e *' have been answered, more than 5868 students have viewed full stepbystep solutions from this chapter. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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 ().

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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