Trigonometric functions of matrices

Template:Use American English Template:Short description The trigonometric functions (especially sine and cosine) for complex square matrices occur in solutions of second-order systems of differential equations. They are defined by the same Taylor series that hold for the trigonometric functions of complex numbers:

$\begin{array}{ccc}\sin X& =X-\frac{X^3}{3!}+\frac{X^5}{5!}-\frac{X^7}{7!}+\cdots & =\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1)!}{X}^{2n+1}\\ \cos X& =I-\frac{X^2}{2!}+\frac{X^4}{4!}-\frac{X^6}{6!}+\cdots & =\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n)!}{X}^{2n}\end{array}$

with Template:Math being the Template:Mvarth power of the matrix Template:Mvar, and Template:Mvar being the identity matrix of appropriate dimensions.

Equivalently, they can be defined using the matrix exponential along with the matrix equivalent of Euler's formula, Template:Math, yielding

$\begin{array}{cc}\sin X& =\frac{e^{iX}-e^{-iX}}{2i}\\ \cos X& =\frac{e^{iX}+e^{-iX}}{2}.\end{array}$

For example, taking Template:Mvar to be a standard Pauli matrix,

${\sigma }_1={\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),$

one has

$\sin (\theta {\sigma }_1)=\sin (\theta ){\sigma }_1,\cos (\theta {\sigma }_1)=\cos (\theta )I,$

as well as, for the cardinal sine function,

$\mathrm{sinc}(\theta {\sigma }_1)=\mathrm{sinc}(\theta )I.$

Properties

The analog of the Pythagorean trigonometric identity holds:

${\sin }^{2}X+{\cos }^{2}X=I$

If Template:Mvar is a diagonal matrix, Template:Math and Template:Math are also diagonal matrices with Template:Math and Template:Math, that is, they can be calculated by simply taking the sines or cosines of the matrices's diagonal components.

The analogs of the trigonometric addition formulas are true if and only if Template:Mvar:

$\begin{array}{c}\sin (X\pm Y)=\sin X\cos Y\pm \cos X\sin Y\\ \cos (X\pm Y)=\cos X\cos Y\mp \sin X\sin Y\end{array}$

Other functions

The tangent, as well as inverse trigonometric functions, hyperbolic and inverse hyperbolic functions have also been defined for matrices:

$\arcsin X=-i\ln (iX+\sqrt{I-X^2})$ (see Inverse trigonometric functions#Logarithmic forms, Matrix logarithm, Square root of a matrix)

$\begin{array}{cc}\sinh X& =\frac{e^X-e^{-X}}{2}\\ \cosh X& =\frac{e^X+e^{-X}}{2}\end{array}$

and so on.