Matrix function
How do you calculate a matrix function in Sage? For example, exp is easily calculated as exp(A) or A.exp(). And how do you calculate sin(A), сos(A), ln(A)?
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3 Answers
It turns out that Sage's various interpreters do have ways to get matrix trig functions :
giac has matrix trig functions :
sage: matrix(giac.sin(A)._sage_())
[ sin(2) -sin(2) + sin(1) -3*sin(2) + 3*sin(1)]
[-3*sin(2) + 3*sin(1) 3*sin(2) - 2*sin(1) 9*sin(2) - 9*sin(1)]
[ sin(2) - sin(1) -sin(2) + sin(1) -3*sin(2) + 4*sin(1)]
sympy has not matrix trig functions. But it has matrix exponential (and logarithm), which can be used to find matrix trigonoimetric functions, as noted by @Max Alekseyev :
sage: matrix([[u for u in v] for v in ((I*A)._sympy_().exp()-(-I*A)._sympy_().ex
....: p())._sage_()/2/I]).apply_map(lambda u:u.demoivre(force=True))
[ sin(2) -sin(2) + sin(1) -3*sin(2) + 3*sin(1)]
[-3*sin(2) + 3*sin(1) 3*sin(2) - 2*sin(1) 9*sin(2) - 9*sin(1)]
[ sin(2) - sin(1) -sin(2) + sin(1) -3*sin(2) + 4*sin(1)]
You can cheat and use Mathematica, or the (gratis but not free) Wolfram engine :
sage: matrix(sin._mathematica_().MatrixFunction(A).sage())
[ sin(2) -sin(2) + sin(1) -3*sin(2) + 3*sin(1)]
[-3*sin(2) + 3*sin(1) 3*sin(2) - 2*sin(1) 9*sin(2) - 9*sin(1)]
[ sin(2) - sin(1) -sin(2) + sin(1) -3*sin(2) + 4*sin(1)]
One notes that conversions of matrices to/from interpreters may be problematic which implies some workarounds in the previous examples...
HTH,
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You can define trigonometric functions via exp() as:
matsin = lambda M: ((I*M).exp() - (-I*M).exp())/2/I
matcos = lambda M: ((I*M).exp() + (-I*M).exp())/2
For logarithm, please see this Q&A: https://math.stackexchange.com/q/3116315
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It's a known issue with Maxima - see https://ask.sagemath.org/question/23784/
A workaround is to define A over CDF, or change its ring on fly:
matcos(A.change_ring(CDF))
Thanks, that helps, although it gives an approximate result. Is there no way to get an exact solution in Sage? The matrix has integer eigenvalues and sin(A) = {{Sin[2], Sin[1] - Sin[2], 3 Sin[1] - 3 Sin[2]}, {3 Sin[1] - 3 Sin[2], -2 Sin[1] + 3 Sin[2], -9 Sin[1] + 9 Sin[2]}, {-Sin[1] + Sin[2], Sin[1] - Sin[2], 4 Sin[1] - 3 Sin[2]}} (computed in Wolfram).
FWIW, sympy has matrix exponential and logarithm (but no matrix trig...). These can be used by Sage :
sage: A = matrix(QQ, 3, [2, -1, -3, -3, 4, 9, 1, -1, -2]) ; A
[ 2 -1 -3]
[-3 4 9]
[ 1 -1 -2]
sage: (((I*A)._sympy_().exp()-(-I*A)._sympy_().exp())._sage_()/2/I).apply_map(lambda u:u.demoivre(force=True))
[ sin(2) -sin(2) + sin(1) -3*sin(2) + 3*sin(1)]
[-3*sin(2) + 3*sin(1) 3*sin(2) - 2*sin(1) 9*sin(2) - 9*sin(1)]
[ sin(2) - sin(1) -sin(2) + sin(1) -3*sin(2) + 4*sin(1)]
Mapping the function to the matrix can be useful:
A = matrix([[0.1,0.2],[0.3,0.4]])
matsin(x)=sin(x)
matlog(x)=log(x)
print(A.apply_map(matsin))
print(A.apply_map(matlog))
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Asked: 2024-05-17 00:20:50 +0100
Seen: 414 times
Last updated: May 19
