Differential Equations and Linear Algebra, 4th Edition

8 Matrix Exponential Methods

8.1 Matrix Exponentials and Linear Systems

The solution vectors of an $n\times n$ homogeneous linear system

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

can be used to construct a square matrix $\mathbf{\text{X}}=\mathbf{\text{Φ}}(t)$ that satisfies the matrix differential equation

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

associated with Eq. (1). Suppose that ${\mathbf{\text{x}}}_1(t),{\mathbf{\text{x}}}_2(t),\dots ,{\mathbf{\text{x}}}_n(t)$ are n linearly independent solutions of Eq. (1). Then the $n\times n$ matrix

$$\mathbf{\text{Φ}}(t)=\left[\begin{array}{ccc}|& |& |\\ {\mathbf{\text{x}}}_1(t)& {\mathbf{\text{x}}}_2(t)& \cdots & {\mathbf{\text{x}}}_n(t)\\ \mathbf{|}& \mathbf{|}& \mathbf{|}\end{array}\right]$$ (2)

having these solution vectors as its column vectors, is called a fundamental matrix for the system in (1).

Fundamental Matrix Solutions