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SciPy Reference Guide, Release 0.8.dev
Trigonometric functions
The trigonometric functions sin , cos , and tan are implemented for matrices in linalg.sinm, linalg.cosm,
and linalg.tanm respectively. The matrix sin and cosine can be defined using Euler’s identity as
The tangent is
and so the matrix tangent is defined as
Hyperbolic trigonometric functions
tan (x) =
sin (A) = ejA − e −jA
2j
cos (A) = ejA + e −jA
sin (x)
cos (x) = [cos (x)]−1 sin (x)
2
[cos (A)] −1 sin (A) .
The hyperbolic trigonemetric functions sinh , cosh , and tanh can also be defined for matrices using the familiar
definitions:
sinh (A) = eA − e −A
2
cosh (A) = eA + e −A
2
tanh (A) = [cosh (A)] −1 sinh (A) .
These matrix functions can be found using linalg.sinhm, linalg.coshm , and linalg.tanhm.
Arbitrary function
Finally, any arbitrary function that takes one complex number and returns a complex number can be called as a matrix
function using the command linalg.funm. This command takes the matrix and an arbitrary Python function. It
then implements an algorithm from Golub and Van Loan’s book “Matrix Computations “to compute function applied
to the matrix using a Schur decomposition. Note that the function needs to accept complex numbers as input in order
to work with this algorithm. For example the following code computes the zeroth-order Bessel function applied to a
matrix.
>>> from scipy import special, random, linalg
>>> A = random.rand(3,3)
>>> B = linalg.funm(A,lambda x: special.jv(0,x))
>>> print A
[[ 0.72578091 0.34105276 0.79570345]
[ 0.65767207 0.73855618 0.541453 ]
[ 0.78397086 0.68043507 0.4837898 ]]
>>> print B
[[ 0.72599893 -0.20545711 -0.22721101]
[-0.27426769 0.77255139 -0.23422637]
[-0.27612103 -0.21754832 0.7556849 ]]
>>> print linalg.eigvals(A)
[ 1.91262611+0.j 0.21846476+0.j -0.18296399+0.j]
>>> print special.jv(0, linalg.eigvals(A))
[ 0.27448286+0.j 0.98810383+0.j 0.99164854+0.j]
>>> print linalg.eigvals(B)
[ 0.27448286+0.j 0.98810383+0.j 0.99164854+0.j]
48 Chapter 1. SciPy Tutorial
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