I want to estimate a matrix $\kappa$ in the equation which is stated below.
$$ B^{-1}X B= \exp(- \kappa t) = N \exp(-D t) N^{-1} $$
There is given that $\kappa$ can be diagnonalized, and further we have that $B$ and $X$ are of dimension $2\times2.$ These two matrices I already have estimated, so the matrix on the left-hand-side in the equation is known. Further it is given that $\kappa=NDN^{-1}$.
How can I solve for $\kappa$ in this situation? As the matrix X is not diagonal, I see this problem as solving $\tilde{X}=Nexp(-Dt)N^{-1}$ where $\tilde{X}$ is a $2 \times 2$ matrix and we want to know $N$ and $D$. Has anybody an idea how to do this?