Comment #1 on issue 3119 by someb...@bluewin.ch: Matrix Exponential
http://code.google.com/p/sympy/issues/detail?id=3119
For the special but very important case of an arbitrary 2x2 matrix we have
a simple formula. (Of course obtained by eigenvalue decomposition) I
implemented this
several times.
# M the original 2x2 matrix
a = M[0,0]
b = M[0,1]
c = M[1,0]
d = M[1,1]
D = sympy.sqrt((a-d)**2 + 4*b*c)/2
t = sympy.exp((a+d)/2)
M = sympy.Matrix([[0,0],[0,0]])
try:
D = sympy.simplify(D)
t = sympy.simplify(t)
except:
pass
if sympy.Eq(D,0):
# special case
M[0,0] = t * (1 + (a-d)/2)
M[0,1] = t * b
M[1,0] = t * c
M[1,1] = t * (1 - (a-d)/2)
else:
# general case
M[0,0] = t * (sympy.cosh(D) + (a-d)/2 * sympy.sinh(D)/D)
M[0,1] = t * (b * sympy.sinh(D)/D)
M[1,0] = t * (c * sympy.sinh(D)/D)
M[1,1] = t * (sympy.cosh(D) - (a-d)/2 * sympy.sinh(D)/D)
# M is now the result of exp(M)
There is also an old paper titled "Some explicit formulas for the matrix
exponential"
by Dennis Bernstein and Wasin So. They show some formulas for the 3x3 case
but only
for special cases.
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