A "Matrix Trigonometry" $e^X$ for matrix $X$ is defined as an always-converging taylor series (provided that $X$ is a $n \times n $ complex matrix):
$$e^X:=\sum_{k=0}^{\infty}\frac{X^k}{k!} $$
A thought occurred to me that we might as well define $\cos(X):=\frac12(e^{iX}+e^{-iX})$ and $\sin(X):=\frac1{2i}(e^{iX}-e^{-iX})$. Now some obvious questions arise:


*

*Is there a generalization for $2\pi$, the period of sine and cosine? Perhaps the best way to do so is to generalize the Euler's Identity $e^{2i\pi}=1$; Is there matrix $T$ such that $e^T=1$? This implies that $\cos (X+T)=\cos (X), \sin(X+T)=\sin (X)$.

*A simple calculation shows that $\cos^2(X)+\sin^2(X)=I$. Can we generalize other trigonometric identities any further?

*Can this concept be used further to derive some useful results? My senses tell me this should find its place in applied mathematics.


If there's any previous reference (which I think is likely) please inform me.
 A: *

*Yes, any diagonalizable matrix $T$ whose eigenvalues are integer multiples of $2 \pi i$ has this property, but this does not imply the identity you want in general unless $X$ and $T$ commute. So in particular $T$ can be scalar. 

*Yes, any trigonometric identity which is a consequence of a polynomial identity between expressions of the form $e^{ix}$ holds also for commuting matrices.

*Really the useful operation is the matrix exponential (solving differential equations, relating Lie algebras and Lie groups, etc.) and everything else is just derived from it. But a keyword you might be interested in is "functional calculus." 
A: The holomorphic functional calculus provides an algebra homomorphism from functions analytic in a neighbourhood of $\sigma(X)$ (the spectrum of $X$) to the closed subalgebra of the $n \times n$ matrices generated by $X$ (more generally, this is true in any complex Banach algebra). 
One way to do this is $$f(X) = \frac{1}{2\pi i} \oint_\Gamma f(z) (zI - X)^{-1}\ dz$$ where $\Gamma$ is a contour that surrounds $\sigma(X)$ in an open set where $f$ is analytic.
 All identities true for ordinary analytic functions carry over to functions of $X$.  
In particular, if we choose a locally constant $f$ whose values in a neighbourhood of $\sigma(X)$ are integer multiples of 
$2 \pi i$, then $f(X)$ satisfies $e^{f(X)} = I$, $\cos(X + f(X)) = \cos(X)$, etc.
