Think of sympy as a tool not a mathematician ;-) *You* know they are
equal, but sympy doesn't go through contortions to figure out if two
things are equal or not when they don't have the same structure. Your
question is answered as FAQ 1 at http://tinyurl.com/nwa87j where it
reminds us that
"The equality operator (==) tests whether expressions have identical
form, not whether they are mathematically equivalent."
So, for example, consider the following mathematical failure
>>> var('x y')
(x, y)
>>> x*(1+y)==x+x*y
False
To sympy, it looks like (at the highest level) that a Mul is being
compared to an Add. So they aren't in the same form.
If you generate expresions which are taking on different forms for the
same quantities, you might get by with a simple expansion to show that
they are equal:
>>> a,b = x*(1+y),x+x*y
>>> a==b
False
>>> a.expand() == b.expand()
True
>>> (a-b)==0
False
>>> (a-b).expand()==0
True
>>>
Looking at your matrix entries you will see:
>>> sa=list(iter(Sigma))
>>> st=list(iter(Sigma.transpose()))
>>> sa[0]
Lambda_00*(Lambda_00 + Lambda_01*Upsilon_01) + Lambda_01*(Lambda_01 +
Lambda_00*Upsilon_01)
>>> st[0]
Lambda_00*(Lambda_00 + Lambda_01*Upsilon_01) + Lambda_01*(Lambda_01 +
Lambda_00*Upsilon_01)
>>> sa[0]==st[0]
True
Well that make sense...it's the corner which doesn't get moved in the
transpose. It's the non-diagonals that are the problem:
>>> print sa[1]
Lambda_10*(Lambda_00 + Lambda_01*Upsilon_01) + Lambda_11*(Lambda_01 +
Lambda_00*Upsilon_01)
>>> print st[1]
Lambda_00*(Lambda_10 + Lambda_11*Upsilon_01) + Lambda_01*(Lambda_11 +
Lambda_10*Upsilon_01)
>>> sa[1]==st[1]
False
Solution? Same as the simple example above:
>>> (sa[1]-st[1]).expand()==0
True
And this suggests a little helper function knowing that a simple
expansion will do:
>>> def mat_eq(a, b):
... z = zip(iter(a), iter(b))
... for zi in z:
... if (zi[0]-zi[1]).expand():
... return False
... else:
... return True
...
>>> mat_eq(Sigma, Sigma.transpose())
True
HTH,
/chris
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