This chapter investigates aspects of two-dimensional linear systems that are not considered in the previous chapter. Some of the topics discussed in Chap. 12 raise questions that are addressed in this chapter. For instance, \(e^{at}c\) is a solution of the differential equation \(x'=ax\) for every real number c. So, is \(e^{At}\mathbf {v}\) a solution of the vector differential equation \(\displaystyle \mathbf {x}'=A\mathbf {x} \qquad \hbox{(1)} \) for every \(2 \times 1\) constant vector \(\mathbf {v}\) , where A is a constant matrix of order 2? But hold on, A is a matrix; so what would \(e^{At}\) even mean? Supposing it could be defined in a way that is natural and compatible with the definition of its scalar counterpart \(e^{at}\) , there is then the question as to its differentiability. If it does have a derivative, is it \(Ae^{At}\) ? Is there a variation of parameter formula for \(\displaystyle \mathbf {x}' = A\mathbf {x} + \mathbf {b}(t) \qquad \hbox{(2)} \) that generalizes the one for nonhomogeneous first-order linear scalar equations? Answers to these and related questions begin in Sects. 13.1 and 13.2 with new ideas and definitions, most notably the fundamental matrix solutions known as principal matrix solutions. Differentiation rules for matrix functions with differentiable entries are stated and proved, after which these rules are used to prove properties of the principal matrix solution denoted by \(Z(t)\) , which is defined to be the matrix function whose columns are linearly independent for all values of t and which satisfies the following matrix differential equation and initial condition: \(\displaystyle \frac {d}{dt}X(t) = AX(t), \quad X(0) =I, \qquad \hbox{(3)} \) where I is the identity matrix. Rather than defining \(e^{At}\) by a power series, as is commonly done, it is defined to be the principal matrix solution \(Z(t)\) , that is, \(e^{At} := Z(t)\) . Consequently, the properties of \(e^{At}\) are the same as those of \(Z(t)\) . In Sect. 13.3, it is shown using a similarity transformation that every \(2 \times 2\) matrix A can be changed into a matrix J called a Jordan (canonical) matrix, of which there are only four types. Several examples using one of the main results in this chapter (Theorem 13.3.4) show how to use Jordan matrices to find solutions of (1). In Sect. 13.4, these matrices also aid in finding the power series expansion of \(e^{At}\) . In Sect. 13.5, phase portraits of (1) for several different matrices A and of the corresponding canonical linear systems \(\mathbf {y}'=J\mathbf {y}\) are obtained by finding formulas for the trajectories of these systems near the equilibrium point \((0,0)\) and then graphing them. The chapter concludes with the derivation of the variation of parameters formula \(\displaystyle \mathbf {x}(t) = e^{A(t - t_0)}{\mathbf {x}}_0 + \int _{t_0}^t e^{A(t - s)}\mathbf {b}(s)\,ds. \qquad \hbox{(4)} \) Several examples and an application illustrate how to use this formula to obtain a solution of a nonhomogeneous equation (2) so that it satisfies a given initial condition.

Matrix Exponential Functions and Series

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发表日期：

2026年

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Leigh C. Becker

摘要

This chapter investigates aspects of two-dimensional linear systems that are not considered in the previous chapter. Some of the topics discussed in Chap. 12 raise questions that are addressed in this chapter. For instance, \(e^{at}c\) is a solution of the differential equation \(x'=ax\) for every real number c. So, is \(e^{At}\mathbf {v}\) a solution of the vector differential equation \(\displaystyle \mathbf {x}'=A\mathbf {x} \qquad \hbox{(1)} \) for every \(2 \times 1\) constant vector \(\mathbf {v}\) , where A is a constant matrix of order 2? But hold on, A is a matrix; so what would \(e^{At}\) even mean? Supposing it could be defined in a way that is natural and compatible with the definition of its scalar counterpart \(e^{at}\) , there is then the question as to its differentiability. If it does have a derivative, is it \(Ae^{At}\) ? Is there a variation of parameter formula for \(\displaystyle \mathbf {x}' = A\mathbf {x} + \mathbf {b}(t) \qquad \hbox{(2)} \) that generalizes the one for nonhomogeneous first-order linear scalar equations? Answers to these and related questions begin in Sects. 13.1 and 13.2 with new ideas and definitions, most notably the fundamental matrix solutions known as principal matrix solutions. Differentiation rules for matrix functions with differentiable entries are stated and proved, after which these rules are used to prove properties of the principal matrix solution denoted by \(Z(t)\) , which is defined to be the matrix function whose columns are linearly independent for all values of t and which satisfies the following matrix differential equation and initial condition: \(\displaystyle \frac {d}{dt}X(t) = AX(t), \quad X(0) =I, \qquad \hbox{(3)} \) where I is the identity matrix. Rather than defining \(e^{At}\) by a power series, as is commonly done, it is defined to be the principal matrix solution \(Z(t)\) , that is, \(e^{At} := Z(t)\) . Consequently, the properties of \(e^{At}\) are the same as those of \(Z(t)\) . In Sect. 13.3, it is shown using a similarity transformation that every \(2 \times 2\) matrix A can be changed into a matrix J called a Jordan (canonical) matrix, of which there are only four types. Several examples using one of the main results in this chapter (Theorem 13.3.4) show how to use Jordan matrices to find solutions of (1). In Sect. 13.4, these matrices also aid in finding the power series expansion of \(e^{At}\) . In Sect. 13.5, phase portraits of (1) for several different matrices A and of the corresponding canonical linear systems \(\mathbf {y}'=J\mathbf {y}\) are obtained by finding formulas for the trajectories of these systems near the equilibrium point \((0,0)\) and then graphing them. The chapter concludes with the derivation of the variation of parameters formula \(\displaystyle \mathbf {x}(t) = e^{A(t - t_0)}{\mathbf {x}}_0 + \int _{t_0}^t e^{A(t - s)}\mathbf {b}(s)\,ds. \qquad \hbox{(4)} \) Several examples and an application illustrate how to use this formula to obtain a solution of a nonhomogeneous equation (2) so that it satisfies a given initial condition.