Differential Equations and Linear Algebra, 4th Edition

7.4 A Gallery of Solution Curves of Linear Systems

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

$${\mathbf{\text{x}}}^{\text{′}}=\mathbf{\text{A}}\mathbf{\text{x}}.$$ (1)

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

$$\mathbf{\text{x}}(t)=\mathbf{\text{v}}{e}^{\lambda t}$$ (2)

is a nontrivial solution of the system (1). Moreover, if A has n linearly independent eigenvectors ${\mathbf{\text{v}}}_1,{\mathbf{\text{v}}}_2,\dots ,$ ${\mathbf{\text{v}}}_n$ associated with its n eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,$ ${\lambda }_n,$ then in fact all solutions of the system (1) are given by linear combinations

$$\mathbf{\text{x}}(t)=c_1{\mathbf{\text{v}}}_1{e}^{{\lambda }_{1}t}+c_2{\mathbf{\text{v}}}_2{e}^{{\lambda }_{2}t}+\cdots +c_n{\mathbf{\text{v}}}_n{e}^{{\lambda }_{n}t},$$ (3)

where $c_1,c_2,\dots ,c_n$ are arbitrary constants. If the eigenvalues include complex conjugate ...