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 equations.[1] They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers:[2]
with Xn being the nth power of the matrix X, and 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, eiX = cos X + i sin X, yielding
For example, taking X to be a standard Pauli matrix,
one has
as well as, for the cardinal sine function,
See also: Axis–angle representation § Exponential map from so(3) to SO(3)
^Gareth I. Hargreaves; Nicholas J. Higham (2005). "Efficient Algorithms for the Matrix Cosine and Sine" (PDF). Numerical Analysis Report. 40 (461). Manchester Centre for Computational Mathematics: 383. Bibcode:2005NuAlg..40..383H. doi:10.1007/s11075-005-8141-0. S2CID 1242875.
^Nicholas J. Higham (2008). Functions of matrices: theory and computation. pp. 287f. ISBN 978-0-89871-777-8.
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