Trigonometric functions of matrices: Difference between revisions

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{{Use American English|date = March 2019}}

The trigonometric functions (especially [[sine]] and [[cosine]]) for real or complex [[square matrices]] occur in solutions of second-order systems of [[differential equation]]s.<ref>{{cite journal|title=Efficient Algorithms for the Matrix Cosine and Sine|authors=Gareth I. Hargreaves, Nicholas J. Higham|journal=Numerical Analysis Report|issue=461|publisher=Manchester Centre for Computational Mathematics|year=2005}}</ref> They are defined by the same [[Taylor series]] that hold for the trigonometric functions of real and [[complex numbers]]:<ref name="Higham">{{cite book|title=Functions of matrices: theory and computation|author=Nicholas J. Higham|year=2008|pages=287f|isbn=9780898717778}}</ref>

{{Short description|Important functions in solving differential equations}}

The '''trigonometric functions''' (especially [[sine]] and [[cosine]]) for real or complex [[square matrices]] occur in solutions of second-order systems of [[differential equation]]s.<ref>{{cite journal|title=Efficient Algorithms for the Matrix Cosine and Sine|authorsauthor=Gareth I. Hargreaves, |author2=Nicholas J. Higham |journal=Numerical Analysis Report|issue=461|publisher=Manchester Centre for Computational Mathematics|year=2005|volume=40|page=383|doi=10.1007/s11075-005-8141-0|bibcode=2005NuAlg..40..383H|s2cid=1242875|url=http://eprints.maths.manchester.ac.uk/124/1/paper2.pdf}}</ref> They are defined by the same [[Taylor series]] that hold for the trigonometric functions of real and [[complex numbers]]:<ref name="Higham">{{cite book|title=Functions of matrices: theory and computation|author=Nicholas J. Higham|year=2008|pages=287f|isbn=9780898717778978-0-89871-777-8}}</ref>

:<math>\begin{align}

\begin{align}

\sin X & = X - \frac{X^3}{3!} + \frac{X^5}{5!} - \frac{X^7}{7!} + \cdots & = \sum_{n=0}^\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}^\infty \frac{(-1)^n}{(2n)!}X^{2n}

\end{align}</math>

with <{{math>|''X^<sup>n</mathsup>''}} being the {{mvar|n}}-th [[Matrix multiplication#Powers of matricesa matrix|power]] of the matrix {{mvar|X}}, and {{mvar|I}} being the [[identity matrix]] of appropriate dimensions.

</math>

with <math>X^n</math> being the {{mvar|n}}-th [[Matrix multiplication#Powers of matrices|power]] of the matrix {{mvar|X}}, and {{mvar|I}} 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]], <{{math>|''e^{^iX}'' {{=}} \cos ''X'' + ''i'' \sin ''X</math>''}}, yielding

:<math>\begin{align}

\begin{align}

\sin X & = {e^{iX} - e^{-iX} \over 2i} \\

\cos X & = {e^{iX} + e^{-iX} \over 2}.

\end{align}</math>

</math>

For example, taking {{mvar|X}} to be a standard [[Pauli matrices|Pauli matrix]],

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one has

:<math>

\sin(\theta \sigma_1) = \sin(\theta)~ \sigma_1 , \qquad \cos (\theta \sigma_1) = \cos (\theta)~I~.,

</math>

as well as, for the [[Sinc function|cardinal sine function]],

:<math>\operatorname{sinc}( \theta \sigma_1) =\operatorname{sinc}( \theta) ~I. </math>

{{see also| Axis–angle representation # Exponential map from so(3) to SO(3)}}

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==Properties==

The analog of the [[Pythagorean trigonometric identity]] holds:<ref name="Higham" />

:<math>\sin^2 X + \cos^2 X = I</math>

:<math>

\sin^2 X + \cos^2 X = I

</math>

If <math>{{mvar|X</math>}} is a [[diagonal matrix]], <{{math>\|sin ''X</math>''}} and <{{math>\|cos ''X</math>''}} are also diagonal matrices with <{{math>|(\sin ''X'')_{_''nn}'' {{=}} \sin( X_{''X<sub>nn})</mathsub>'')}} and <{{math>|(\cos ''X'')_{_''nn}'' {{=}} \cos (X_{''X<sub>nn})</mathsub>'')}}, that is, they can be calculated by simply taking the sines or cosines of the matricematrices's diagonal components.

The analogs of the [[trigonometric addition formulas]] are true [[if and only if]] <math>{{mvar|XY {{=}} YX</math>}}:<ref name="Higham" />

:<math>\begin{align}

\begin{align}

\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{align}</math>

</math>

==Other functions==

The tangent, as well as [[inverse trigonometric functions]], [[hyperbolic function|hyperbolic]] and [[inverse hyperbolic function]]s have also been defined for matrices:<ref>[https://help.scilab.org/docs/5.5.2/en_US/section_99038107015b1d789de50bf92f154a85.html Scilab trigonometry].</ref>

:<math>\arcsin X = -i \ln \left( iX + \sqrt{I-X^2} \right)</math> (see [[Inverse trigonometric functions#Logarithmic forms]], [[Matrix logarithm]], [[Square root of a matrix]])

:<math>

:<math>\begin{align}

\arcsin X = -i \ln \left( iX + \sqrt{I-X^2} \right)</math> (see [[Inverse trigonometric functions#Logarithmic forms]], [[Matrix logarithm]], [[Square root of a matrix]])

:<math>

\begin{align}

\sinh X & = {e^X - e^{-X} \over 2} \\

\cosh X & = {e^X + e^{-X} \over 2}

\end{align}</math>

and so on.

</math>

etc.

==References==

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[[Category:Trigonometry]]

[[Category:Matrix theory]]

{{math-stub}}