$$x_{k+2} - (2b) x_{k+1}+x_k=0$$ Here $x_k$ is a $2 \times 1$ column matrix sequence ($ k \ge 1$), $b$ is a constant.
I know the form of answer, $$x_k=A \cos(k\cos^{-1}(b))+B\sin(k\cos^{-1}(b))$$ Here $A,B$ are $2 \times 1$ constant matrices. However I cannot derive the answer.
Usually, in similar case, I use a form of recursion formula $(x_{k+2}-a \cdot x_{k+1})=c(x_{k+1}-a \cdot x_k)$ But I'm not accustomed to matrix recurrence relation and this form doesn't work well in this case... And I think in this way, I cannot find the answer in given form. Can somebody help me?
