- 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 8-1 1, determine whether the given matrix is a fundament...
- 3.1 1.9: In each of 8-1 1, determine whether the given matrix is a fundament...
- 3.1 1.10: In each of 8-1 1, determine whether the given matrix is a fundament...
- 3.1 1.11: In each of 8-1 1, determine whether the given matrix is a fundament...
- 3.1 1.12: Let +'(t) be the solution of the initial-value 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 AB-BA. (...
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
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!
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.
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.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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).
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).
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.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , 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. A-I is also symmetric.