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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.

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  • $\begingroup$ I get three distinct eigenvalues, why wouldnt this be diagonalizable? $\endgroup$ – Olba12 Sep 20 '18 at 0:34
  • $\begingroup$ @Olba12 I don't claim it's not diagonalizable in a different basis. It's not diagnolized in its original form. Do you think diagonalization may be of help? Could you please describe your approach? $\endgroup$ – Swistack Sep 20 '18 at 0:36
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    $\begingroup$ If $A=PDP^{-1}$ is diagonalizable then $$f(A) = P \begin{bmatrix} f(d_1) & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & f(d_n) \end{bmatrix}P^{-1}$$ $\endgroup$ – Olba12 Sep 20 '18 at 0:38
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    $\begingroup$ The matrix built by eigenvectors to corrensponding eigenvalues $\endgroup$ – Olba12 Sep 20 '18 at 0:41
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    $\begingroup$ Swistack, check the numbers in the preparation paper against what you typed above. You cannot expect to do this problem unless the eigenvalues are very easy, indeed integers. Then $\sin W = (e^{iW} - e^{-iW})/(2i)$ and you can find the exponential parts if you can explicitly diagonalize $A$ $\endgroup$ – Will Jagy Sep 20 '18 at 0:56
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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).

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