solving matrix equations
mb
Say I want to compute the centralizer of a matrix.
sage: P.<a,b,c,d>=PolynomialRing(QQ)
sage: A=matrix(P,2,2,[2,1,1,1])
sage: B=matrix(P,2,2,[a,b,c,d])
sage: C=A*B-B*A
sage: C
[ -b + c -a + b + d]
[ a - c - d b - c]
sage: var('a b c d')
(a, b, c, d)
sage: solve([-b + c==0,-a + b + d==0,a - c - d==0,b - c==0],[a,b,c,d])
[[a == r2 + r1, b == r2, c == r2, d == r1]]
Here I had to manually copy the entries of C into solve, which is a
problem for larger matrices. Ideally, something like
solve(C==0,[a,b,c,d])
should work, but of course it doesn't. Is there any way of doing this?
Mladen
Craig Citro
Yep, there are definitely easy ways of doing this. Here's one way:
sage: var('a b c d')
(a, b, c, d)
sage: B = matrix(2,[a,b,c,d])
sage: C = A*B - B*A
sage: [ e == 0 for e in C.list() ]
[c - b == 0, d + b - a == 0, -d - c + a == 0, b - c == 0]
sage: solve([ e == 0 for e in C.list() ], N.list())
Jason Grout
> Hi,
>
> Yep, there are definitely easy ways of doing this. Here's one way:
>
> sage: var('a b c d')
> (a, b, c, d)
> sage: A = matrix(2,[2,1,1,1])
> sage: B = matrix(2,[a,b,c,d])
> sage: C = A*B - B*A
>
> sage: [ e == 0 for e in C.list() ]
> [c - b == 0, d + b - a == 0, -d - c + a == 0, b - c == 0]
>
> sage: solve([ e == 0 for e in C.list() ], N.list())
> [[a == r2 + r1, b == r2, c == r2, d == r1]]
Just to point out for those that need a guide through the above, Craig
is basically creating a list of equations from each entry in the matrix.
He doesn't show it, but I suppose that N is a matrix of the variables.
You could also do:
solve([ e == 0 for e in C.list() ], C.variables())
That said, I think it would be a great thing if solve could recognize
matrices and that two matrices are equal if each entry is equal. I
believe MMA does this (but it's easier there; matrices are nothing more
than nested lists). It'd certainly make certain things I do more
natural if I could do:
solve(matrixA==matrixB)
and that was equivalent to:
solve([i==j for i,j in zip(matrixA.list(), matrixB.list())])
if the matrices were of the same dimensions.
Okay, so now that I've written my piece, I suppose the next step is to
open a trac ticket, write a patch to implement it, and post it for
review :).
The ticket is http://trac.sagemath.org/sage_trac/ticket/5201
I won't cry if someone submits a patch before I get to it :).
Jason