The exponential matrix

The fundamental matrix $\Omega(t)$ which satisfies the initial condition
$\Omega(0)=I$, the identity matrix, is the exponential matrix

$$\Omega(t)=e^{tA}=\sum_{n=0}^\infty{ (tA)^n\over n!} = I +tA + {t^2 A^2\over 2!}+ ...$$ (7)

If A is diagonalizable then A=PDP^-1 and

   $$\begin{eqnarray} e^{At}=I +tA + {t^2 A^2\over 2!}+ ...\nonumber\ =PP^{-1}+tPDP^... ...dots&\vdots&\vdots&\vdots\cr ...& ...&...& e^{\sigma_n t}} P^{-1}.\end{eqnarray}$$

A solution satisfying a prescribed initial condition X(0) is X(t)=e^AtX(0).

Example 3

 Solve $\dot X=AX$,

$$A=\pmatrix{0&1&1\cr 1&0&1\cr 1&1&0},\quad X(0)=\pmatrix{1\cr 2\cr 3}.$$

We find the eigenvalues to be $\sigma_1=\sigma_2=-1$, with eigenvectors

$$\xi_1=\pmatrix{1\cr 0\cr -1}, \quad \xi_2=\pmatrix{0\cr 1\cr -1},$$

and $\sigma_3=2$ with

$$\xi_3=\pmatrix{1\cr 1\cr 1}.$$

The solution is X(t)=e^AtX(0) where the eigenvectors are used to diagonalize the matrix

$$e^{At}=Pe^{Dt}P^{-1}=P\pmatrix {e^{-t}&0&0\cr 0&e^{-t}&0\cr 0&0&e^{2t}}P^{-1}, \quad P=\pmatrix{1&0&1\cr 0&1&1\cr -1 &-1&1}$$

$$=\pmatrix{1&0&1\cr 0&1&1\cr -1 &-1&1}\pmatrix{e^{-t}&0&0\c... ... {2\over 3}&-{1\over 3}\cr {1\over 3}&{1\over 3}& {1\over 3} }.$$

Nondiagonalizable matrices. Not every matrix A has a basis of eigenvectors so not every matrix can be diagonalized. Every square matrix can be transformed to Jordan canonical form. A Jordan block is a square matrix having the form  

$$J=\pmatrix{\sigma &1 &0&... \cr 0&\sigma& 1& 0...\cr \vdots&\vdots&\vdots&\vdots\cr 0&...&\sigma&1\cr ...& ...&...& \sigma }.$$ (8)

Theorem 1

 If A is a general matrix of order n, there is a nonsingular matrix Q such that

$$Q^{-1}AQ= \pmatrix{J_1 &0 &0&... \cr 0&J_2& 0& 0...\cr \vdots&\vdots&\vdots&\vdots\cr 0&...&0 &J_k },$$

where the J_i are Jordan blocks. The same eigenvalues can occur in different blocks. The number of distinct blocks corresponding to a given eigenvalue is equal to the number of independent eigenvectors corresponding to that eigenvalue.

If A is not diagonalizable, we must distinguish between the algebraic and geometric multiplicity of an eigenvalue. The algebraic multiplicity is the multiplicity as a root of the characteristic polynomial ${\rm det}(A-\sigma I)=0$. The geometric multiplicity is the number of linearly independent eigenvectors. For instance, the matrix

$$A=\pmatrix{1&0\cr 1&1}$$ (9)

has an eigenvalue 1 of algebraic multiplicity 2 but geometric multiplicity 1.

We finally remark that there is nothing magical about the number 1 appearing in the off-diagonal position of the Jordan blocks. We can replace 1 by any other nonzero number $\epsilon$; in particular, we can make $\epsilon$ as small as we please, as long as it is not zero. We shall take advantage of this later when we discuss nonlinear stability.