# [SOLVED]Fundamental Matrix

#### dwsmith

##### Well-known member
Given
$\mathbf{A} = \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & -5 & -4 \end{bmatrix}$
find the fundamental matrix.

If I had real eigenvectors, I could simply do
$e^{\mathbf{A}t} = \mathbf{S}e^{\mathbf{D}t}\mathbf{S}^{-1},$
but I have complex eigenvalues and eigenvectors.

Also, if the $$\mathbf{A}^n$$ shows a pattern, I could do say, for example, $$\mathbf{A}^n = \begin{bmatrix} 2^n & 1\\ 3^n & 0\end{bmatrix}$$, but when I look at $$A^n$$, I have
\begin{align}
\mathbf{A}^2 &= \begin{bmatrix}
0 & 0 & 1\\
0 & -5 & -4\\
0 & 20 & 11
\end{bmatrix}\\
\mathbf{A}^3 &= \begin{bmatrix}
0 & -5 & -4\\
0 & 20 & 11\\
0 & -55 & -24
\end{bmatrix}\\
\mathbf{A}^4 &= \begin{bmatrix}
0 & 20 & 11\\
0 & -55 & -24\\
0 & 120 & 41
\end{bmatrix}
\end{align}

So for the methods I know, I can't find the fundamental matrix. What can I do?