Homogeneous Linear Systems Three Cases: Distinct Real Eigenvalues - - PowerPoint PPT Presentation

Homogeneous Linear Systems

Three Cases: ◮ Distinct Real Eigenvalues ◮ Repeated Eigenvalues ◮ Complex Eigenvalues

MSW MTH 291

Distinct Eigenvalues

General Solution-Homogeneous Systems Let λ1, λ2, . . . , λn be n distinct real eigenvalues of the coefficient matrix A of the homogeneous system X′ = AX and let K1, K2, . . . , Kn be the corresponding

• eigenvectors. Then the general solution of X′ = AX on the interval (−∞, ∞) is given

by X = c1K1eλ1t + c2K2eλ2t + · · · + cnKneλnt (1)

MSW MTH 291

Repeated Eigenvalues

• 1. For some nx n matrices A it may be possible to find m linearly independent

eigenvectors K1, K2 . . . , Km corresponding to an eigenvalue λ1 of multiplicity m ≤ n. In this case the general solution of the system contains the linear combination c1K1eλ1t + c2K2eλ1t + · · · + cmKmeλ1t (2)

• 2. If there is only one eigenvector corresponding to the eigenvalue λ1 of multiplicity

m, then m linearly independent solutions of the form X1 = K11eλ1t X2 = K21teλ1t + K22eλ1t X3 = K31 t2 2 eλ1t + K32teλ1t + K33eλ1t . . . Xm = Km1 tm−1 (m − 1)! eλ1t + Km2 tm−2 (m − 2)! eλ1t + · · · + Kmmeλ1t where Kij are column vectors, can always be found.

MSW MTH 291

Complex Eigenvalues

Solutions Corresponding to a Complex Eigenvalue Let A be the coefficient matrix having real entries of the homogeneous system X′ = AX, and let K1 be an eigenvector corresponding to the complex eigenvalue λ1 = α + iβ, where α and β are real. Then K1eλ1t and K1eλ1t are solutions of X′ = AX. Real Solutions Corresponding to a Complex Eigenvalue Let λ1 = α + iβ be a complex eigenvalue of the coefficient matrix A in the homogeneous system X′ = AX and let B1 = 1 2 (K1 + K1)

• and B2 = i

2 (−K1 + K1)

• (3)

Re(K1) Im(K1) where B1 and B2 are column vectors. Then X1 = [B1 cos βt − B2 sin βt]eαt X2 = [B2 cos βt + B1 sin βt]eαt (4) are linearly independent solutions of X′ = AX on (−∞, ∞).

MSW MTH 291