7.4 A Gallery of Solution Curves of Linear Systems

In the preceding section we saw that the eigenvalues and eigenvectors of the n×n matrix A are of central importance to the solutions of the homogeneous linear constant-coefficient system

x=Ax. (1)

Indeed, according to Theorem 1 from Section 7.3, if λ is an eigenvalue of A and v is an eigenvector of A associated with λ, then

x(t)=veλt (2)

is a nontrivial solution of the system (1). Moreover, if A has n linearly independent eigenvectors v1, v2, , vn associated with its n eigenvalues λ1, λ2, , λn, then in fact all solutions of the system (1) are given by linear combinations

x(t)=c1v1eλ1t+c2v2eλ2t++cnvneλnt, (3)

where c1,c2,,cn are arbitrary constants. If the eigenvalues include complex conjugate ...

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