Let $A = 2\begin{pmatrix}4 & -1 & -3 \\ -2 & 1 & 1 \\-2 & 3 & -4\end{pmatrix}$
I want to compute $\sin\left(\dfrac{\pi A}{2}\right)$ for this matrix. I know that sinus of a matrix can be expressed as sum of infinite series, but for this I have to know $A^k$. Am I on the right direction or there're other approaches to this problem? Any help would be greatly appreciated.
The general idea is as follows.
Let $A\in M_3(\mathbb{Q})$ that admits $3$ distinct real eigenvalues $(\lambda_i)_i$ and let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Note that $A$ is diagonalizable over $\mathbb{R}$ and $f(A)$ commute with $A$, and consequently, is a polynomial in $A$.
Let $P\in \mathbb{R}[x]$ be the Lagrange interpolating polynomial of degree $2$ that sends the $\lambda_i's$ on the $f(\lambda_i)'s$ ($P(x)=a+bx+cx^2$). Then $f(A)=P(A)=aI_3+bA+cA^2$.
Unfortunately, here, the $(\lambda_i)$ are in an algebraic extension of $\mathbb{Q}$ of degree $6$; we can explicitly calculate the $(\lambda_i)'s$ but the calculations of $a,b,c$ is quasi unfeasible (except for Maple for example).