Differential Equations and Linear Algebra, 4th Edition

7.2 Matrices and Linear Systems

A system of differential equations often can be simplified by expressing it as a single differential equation involving a matrix-valued function. A matrix-valued function, or simply matrix function, is a matrix such as

$$\mathbf{\text{x}}(t)=\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ ⋮\\ x_n(t)\end{array}\right]$$ (1)

or

$$\mathbf{\text{A}}(t)=\left[\begin{array}{cccc}{a}_{11}(t)& a_{12}(t)& \cdots & a_{1n}(t)\\ a_{21}(t)& a_{22}(t)& \cdots & a_{2n}(t)\\ ⋮& ⋮& ⋮\\ a_{m1}(t)& a_{m2}(t)& \cdots & a_{mn}(t)\end{array}\right],$$ (2)

in which each entry is a function of t. We say that the matrix function A(t) is continuous (or differentiable) at a point (or on an interval) if each of its elements has the same property. The derivative of a differentiable matrix function is defined by elementwise differentiation; that is,

$$\mathbf{\text{A}}\text{′}(t)={\displaystyle \frac{d\mathbf{\text{A}}}{dt}}=\left[{\displaystyle \frac{d{a}_{ij}}{dt}}\right].$$ (3)

Example 1

If

$$\mathbf{\text{x}}(t)=\left[\begin{array}{c}t\\ t^2\\ e^{-t}\end{array}\right]\text{and}\mathbf{\text{A}}(t)=\left[\begin{array}{cc}\sin t& 1\\ t& \cos t\end{array}\right],$$

then

$${\displaystyle \frac{d\mathbf{\text{x}}}{dt}}=\left[\begin{array}{c}1\\ 2t\\ -e^{-t}\end{array}\right]\text{and}\mathbf{\text{A}}\text{′}(t)=$$