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[SOLVED] Fundamental Matrix

dwsmith

Well-known member
Feb 1, 2012
1,673
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?