Differential Equations and Linear Algebra, 4th Edition

8.3 Spectral Decomposition Methods

Here, we present an alternative approach to the computation of the matrix exponential $e^{\mathbf{\text{A}}t}$, one that does not require that eigenvectors (including generalized ones) of the $n\times n$ matrix A be found first. Assume that the characteristic polynomial of A is written in the form

$$p(\lambda )={(-1)}^n\left|\mathbf{\text{A}}-\lambda \mathbf{\text{I}}\right|,$$ (1)

with leading term $+{\lambda }^n$. [Compare Eqs. (4) and (5) in Section 6.1.] If the (not necessarily distinct) eigenvalues of A are ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$, then

$$p(\lambda )=(\lambda -{\lambda }_1)(\lambda -{\lambda }_2)\cdots (\lambda -{\lambda }_n).$$ (2)

The Cayley-Hamilton theorem (Section 6.3) says that any matrix A satisfies its own characteristic equation; that is,

$$p(\mathbf{\text{A}})={\displaystyle \prod _{i=1}^n(\mathbf{\text{A}}-{\lambda }_i\mathbf{\text{I}})=\mathbf{0}}$$ (3)

(where I denotes the $n\times n$ identity matrix). This crucial fact is the key to our method in this section.

The way we proceed ...