Let $x'=Ax, A\in \mathbb{R^{n,n}}$, be a linear autonomous system. Denote by $\{ \lambda_j\}$ the set of the eigenvalues of $A$. I want to study its stability. There is this following fact that im supposed to use:
The systen is stable (i.e. all solutions are bounded) if $Re(\lambda _j)\leq 0$ for all $j$ and $g_j=a_j$ if $Re(\lambda _j)=0$.
Where $g_j $ and $a_j$ denote the geometric resp. algebraic multiplicity of $\lambda _j$.
My question is, what if the eigenvectors are complex? How do we "count" the geometric multiplicity then? For example, let $$A=\begin{bmatrix}
0&2&1\\
-2&0&2\\
0&0&-3\\
\end{bmatrix}$$
Its eigenvalues are $(-3,-2i,2i)$ where $a_j=1$ . Now, to know if its stable or not, i need to calculate the geometric multiplicities of the eigenvalues 2i and -2i. What are they? Is this system stable? Do we count each complex eigenvalue as 1 independent vector or 2 independent vectors, one real and the other complex?
Best Answer
Yor matrix $A$ is $3 \times 3$ and $A$ has $3$ different eigenvalues, hence the geometric multiplicities of each eigenvalue $=1.$