You can adapt the power series definition of cosine to take in a matrix. Does this have a geometric interpretation/definition? Can it be used for various purposes? I actually have extended the matrix product to a broad class of manifolds, so I'm wondering how important this advance is.
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$\begingroup$ researchgate.net/profile/Jacob_Wakem $\endgroup$– Insulin69Commented Jun 28, 2022 at 23:48
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1$\begingroup$ I imagine the geometric interpretation would depend on precisely which adaptation you took. For example, if you used the Euler identity as your inspiration, that would be one thing. If you literally just plugged a matrix into $1-x^2/2 + x^4/4! - \cdots$ that would be another. $\endgroup$– Ryan BudneyCommented Jun 29, 2022 at 0:16
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$\begingroup$ @RyanBudney Apparently systems of differential equations can be solved using the cosine of a matrix. Wonder what you could solve with the cosine of a manifold. $\endgroup$– Insulin69Commented Jun 29, 2022 at 0:46
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Maybe something comes from the differential equation ...
Let $A$ be an $n \times n$ matrix. The $C^2$ function
$y : \mathbb R \to \mathbb R^n$ satisfies the differential equation
$$
y''(t)+A^2 \big(y(t)\big)=0, \quad y(0)=y_0, y'(0)=0
$$
if and only if
$$
y(t) = \cos(tA) \big(y_0\big)
$$
answered Jun 29, 2022 at 0:49
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$\begingroup$ As you say, this is a comment (in fact apparently in the same spirit as @Insulin's earlier comment); it does not seem to be an answer. $\endgroup$– LSpiceCommented Jun 29, 2022 at 1:14