I have the following matrix:
$$ \Lambda=\begin{bmatrix} \Lambda_1 \\ & \Lambda_2 \\ & & \ddots \\ & & & \Lambda_m \end{bmatrix}\in\mathbb R^{n\times n} $$
Where $\Lambda_1\in\mathbb C^{2\times 2}$ equals the following:
$$ \Lambda_1=\begin{bmatrix} \sigma+j\omega & 0 \\ 0 & \sigma-j\omega \end{bmatrix} $$
Note that $\Lambda$ is the Jordan decomposition of matrix $A\in\mathbb R^{n\times n}$ with respect to the basis $\{v,v^*,v_3,...,v_n\}$ where $v=x+jy\in\mathbb C^n$ and $v^*=x-jy\in\mathbb C^n$ are two eigenvectors of $A$ associated with $\lambda=\sigma+j\omega$ and $\lambda^*=\sigma-j\omega$ respectively.
Question: I need to find transformation matrix, $T$, into a new basis (specify what basis it is, too) in which:
$$ T\Lambda T^{-1}=\begin{bmatrix} \widetilde\Lambda_1 \\ & \Lambda_2 \\ & & \ddots \\ & & & \Lambda_m \end{bmatrix}\in\mathbb R^{n\times n} $$
Where $\widetilde\Lambda_1\in\mathbb R^{2\times 2}$.
The trouble is, I am unable to find this new basis, nor the transformation matrix for it. My current thinking is that the new basis should be:
$$ \left\{ \begin{bmatrix} \sigma+j\omega \\ 0 \\ \vdots \\ 0 \end{bmatrix}\in\mathbb C^n, \begin{bmatrix} 0 \\ \sigma-j\omega \\ \vdots \\ 0 \end{bmatrix}\in\mathbb C^n,v_3,...,v_n \right\} $$
Which would give $\widetilde A_1=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$, however at that point (assuming the basis is correct), I am unable to find the change of basis matrix... Thanks for helping me out with this!