Differential Equations and Linear Algebra, 4th Edition

7.3 The Eigenvalue Method for Linear Systems

We now introduce a powerful method for constructing the general solution of a homogeneous first-order system with constant coefficients,

$$\begin{array}{rclllllll}{x}_1^{\text{′}}& =& a_{11}{x}_1& +& a_{12}{x}_2& +& \cdots & +& a_{1n}{x}_n,\\ x_2^{\text{′}}& =& a_{21}{x}_1& +& a_{22}{x}_2& +& \cdots & +& a_{2n}{x}_n,\\ ⋮\\ x_n^{\text{′}}& =& a_{n1}{x}_1& +& a_{n2}{x}_2& +& \cdots & +& a_{nn}{x}_{n.}\end{array}$$ (1)

By Theorem 3 of Section 7.2, we know that it suffices to find n linearly independent solution vectors ${\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,\dots ,{\mathbf{\text{x}}}_n$; the linear combination

$$\mathbf{\text{x}}(t)=c_1{\mathbf{\text{x}}}_1+c_2{\mathbf{\text{x}}}_2+\cdots +c_n{\mathbf{\text{x}}}_n$$ (2)

with arbitrary coefficients will then be a general solution of the system in (1).

To search for the n needed linearly independent solution vectors, we proceed by analogy with the characteristic root method for solving a single ...