Linear Systems with Constant Coefficients

Abstract

The bulk of this chapter is devoted to homogeneous linear systems of ODEs with real constant coefficients. This means systems of the form

$$\displaystyle{ \begin{array}{ccc} x_{1}^{{\prime}}& =& a_{11}x_{1} + a_{12}x_{2} +\ldots +a_{1d}x_{d}, \\ x_{2}^{{\prime}}& =& a_{21}x_{1} + a_{22}x_{2} +\ldots +a_{2d}x_{d},\\ \\ \vdots&\vdots&\vdots\\ \\ x_{d}^{{\prime}}& =&a_{d1}x_{1} + a_{d2}x_{2} +\ldots +a_{dd}x_{d}.\end{array} }$$

(2.1)

(From now on, we shall let d be the dimension of our systems, so that the index n is available for other uses.) The written-out system (2.1) is awkward to read or write, and we shall normally use the vastly more compact linear-algebra notation

$$\displaystyle{ \mathbf{x}^{{\prime}} = A\mathbf{x}, }$$

(2.2)

where x = (x ₁, x ₂, …, x _d ) is a d-dimensional vector of unknown functions, A is a d × d matrix with real entries, and matrix multiplication is understood in writing A x. In vector notation, an appropriate initial condition for (2.2) is

$$\displaystyle{ \mathbf{x}(0) = \mathbf{b}, }$$

(2.3)

where $\mathbf{b} \in \mathbb{R}^{d}$.