Fundamental Matrix of \(\mathbf{A}\) - Find the Solution

[Dustinsfl]

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?

 

[jvicens]

I understand your frustration in not being able to find a clear pattern or solution for the fundamental matrix. However, there are a few things you can try to find a solution.

Firstly, you can try using the Jordan canonical form of the matrix \(\mathbf{A}\) instead of the diagonal form. This may help in finding a pattern for \(\mathbf{A}^n\) and ultimately the fundamental matrix.

Secondly, you can try using numerical methods such as power series or Taylor series approximations to find an approximate solution for the fundamental matrix. This may not give an exact solution, but it can be a good starting point for further analysis.

Lastly, you can also try using software such as MATLAB or Mathematica to find the fundamental matrix numerically. These software have built-in functions for solving differential equations and finding fundamental matrices.

In conclusion, while the methods you know may not be applicable in this case, there are still other approaches you can try to find a solution for the fundamental matrix. Keep exploring and experimenting, and you may find a solution that works for your specific problem.