of Systems first order linear equations (1) Fundamental Matrices x1 (t), . . . , xn (t) form a fundamental set of solutions of x = A(t)x. 1 x1 (t) xn (t) 1 (t) = 1 n xn (t) xn (t) Then ,
whose columns are the vectors x1 (t), . . . , xn (t), is a fundamental matrix. A fundamental matrix is nonsingular and = A(t). The general solution of the system is c1 x1 (t) + + cn xn (t) or x = (t)c where c = (c1 , , cn )T . If the initial condition is x(t0 ) = x0 , (t0 )c = x0 and c = -1 (t0 )x0 since (t0 ) is nonsingular. Thus, the solution of the IVP becomes x = (t)-1 (t0 )x0 . The special fundamental matrix (t), whose columns are the vectors x1 (t), . . . , xn (t), with the initial conditions xj (t0 ) = ej , i.e, 1 0 0 0 1 0 (t0 ) = = I; 0 0 1 now the solution of the IVP becomes x = (t)x0 . Note (t) = (t)-1 (t0 ). exp(At) = (t). Transformation into a diagonal matrix: If 1 , . . . , n are all different, and 1 , . . . , n are linearly independent eigenvectors, form the matrix T as 1 n 1 1 T = . n 1 n n det T = 0, AT = T D where D= 1 0 0 0 2 0 0 0 n
is a diagonal matrix whose diagonal entries are the eigenvalues of A. D = T -1 AT : similarity transformation, A: diagonalizable. If A has fewer than n linearly indeoendent eigenvectors, A is not diagonalizable. Consider x = Ax. Define x = T y and then we have y = T -1 AT y = Dy whose fundamental matrix is the diagonal matrix e1 t 0 0 0 e2 t 0 Q(t) = exp(Dt) = . 0 0 en t A fundamental matrix for x = Ax is = T Q.
(2)
Repeated Eigenvalues Suppose r = is a k-fold root of det(A - rI) = 0; is an eigenvalue of algebraic multiplicity k of A. If there are k linearly indenpendent eigenvectors, 1 , . . . , k , then x1 = 1 et , . . . , xk = k et are linearly indenpendent so solutions it makes no difference. If there are fewer than k linearly indenpendent eigenvectors, it is necessary to find other solutions of a different form. Suppose r = is a double eigenvalue of A and there is only one corresponding eigenvector . Then one solution is x1 = et , where (A - I) = 0 and a second solution is x2 = tet + et , where (A - I) = . is a generalized eigenvector. Jordan form: if A has repeated eigenvalues, A can be transformed into a nearly diagonal matrix (Jordan form), which has the eigenvalues of A on the mian diagonal, ones in certain positions on the diagonal above the main diagonal, and zeros elsewhere.
(3)
Nonhomogeneous Linear Systems x = A(t)x + g(t): the general solution x(t) = c1 x1 (t) + + cn xn (t) + v(t). Diagonalization: if A is diagonizable, use T defined above and x = T y. T y = AT y + g(t) y = T -1 AT y + T -1 g(t) = Dy + h(t), a system of n uncoupled equations, yj (t) = j yj (y) + hj (t), j = 1, . . . , n. t Thus, yj (t) = ej t t0 e-j s hj (s)ds + cj ej t , j = 1, . . . , n where cj are arbitrary constants. Undetermined coefficients If ...