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Trigonometric functions of matrices

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The trigonometric functions (especially sine and cosine) for real or 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 real and complex numbers:

Contents

sin X = X X 3 3 ! + X 5 5 ! X 7 7 ! + = n = 0 ( 1 ) n ( 2 n + 1 ) ! X 2 n + 1 cos X = I X 2 2 ! + X 4 4 ! X 6 6 ! + = n = 0 ( 1 ) n ( 2 n ) ! X 2 n

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

sin X = e i X e i X 2 i cos X = e i X + e i X 2 .

For example, taking X to be a standard Pauli matrix,

σ 1 = σ x = ( 0 1 1 0 )   ,

one has

sin ( θ σ 1 ) = sin ( θ )   σ 1 , cos ( θ σ 1 ) = cos ( θ )   I   ,

as well as, for the cardinal sine function,

sinc ( θ σ 1 ) = sinc ( θ )   I .

Properties

The analog of the Pythagorean trigonometric identity holds:

sin 2 X + cos 2 X = I

If X is a diagonal matrix, sin X and cos X are also diagonal matrices with (sin X)nn = sin(Xnn) and (cos X)nn = cos(Xnn), that is, they can be calculated by simply taking the sines or cosines of the matrice's diagonal components.

The analogs of the trigonometric addition formulas are true if and only if XY = YX:

sin ( X ± Y ) = sin X cos Y ± cos X sin Y cos ( X ± Y ) = cos X cos Y sin X sin Y

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 ( i X + I X 2 ) (see Inverse trigonometric functions#Logarithmic forms, Matrix logarithm, Square root of a matrix) sinh X = e X e X 2 cosh X = e X + e X 2

and so on.

References

Trigonometric functions of matrices Wikipedia