How to calculate sine of a matrix

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You could just use

$$ sin(x) = \frac{exp(ix)-exp(-ix)}{2*i} $$

I'm just putting down some thoughts here that hopefully someone else will know about.

sage: M = matrix([[2,3],[3,2]])
sage: e^M
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
TypeError: mutable matrices are unhashable
sage: exp(M)
[1/2*(e^6 + 1)*e^(-1) 1/2*(e^6 - 1)*e^(-1)]
[1/2*(e^6 - 1)*e^(-1) 1/2*(e^6 + 1)*e^(-1)]
sage: M.exp()
[1/2*(e^6 + 1)*e^(-1) 1/2*(e^6 - 1)*e^(-1)]
[1/2*(e^6 - 1)*e^(-1) 1/2*(e^6 + 1)*e^(-1)]

Naturally, this is really because there isn't any sin method. However:

sage: M = matrix([[2,3],[3,2]])
sage: M.simp[tab] # nothing
sage: M = matrix(SR,[[2,3],[3,2]])
sage: M.simp[tab]
M.simplify           M.simplify_rational  M.simplify_trig

There still isn't a sin. But this could be doable. Of course, the question remains as to how many such things we would want to add. Thoughts?

I'm trying to understand what you are doing with sin(Mt). Is one of these what you want?

sage: m=random_matrix(QQ,3); m
[-1  0 -1]
[-1  1  2]
[-2  0  1]
sage: m.apply_map(sin)
[-sin(1)       0 -sin(1)]
[-sin(1)  sin(1)  sin(2)]
[-sin(2)       0  sin(1)]
sage: t=var('t')
sage: (m*t).apply_map(sin)
[  sin(-t)         0   sin(-t)]
[  sin(-t)    sin(t)  sin(2*t)]
[sin(-2*t)         0    sin(t)]

Or do you mean you want to calculate the series $\sum_{k=0}^{\infty} (-1)^{k}\frac{M^{2k+1}}{(2k+1)!}$?