Evaluate polynomial expression on matrices

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How can I construct a map $M_2(\mathbb Q)^3 \to M_2(\mathbb Q), (A,B,C) \mapsto (AB-BA)C$ where $M_2(\mathbb Q)$ denotes the space of 2x2 matrices.

sage: x,y,z = var('x y z')
sage: f=(x*y-y*x)*z
sage: f
0

Comments

Like this:

def f(A,B,C):
        return (A*B-B*A)*C

FrédéricC ( 6 years ago )

To further Frederic's answer :

sage: sage: x,y,z = var('x y z')
....: sage: f=(x*y-y*x)*z
....: sage: f
....: 
0
sage: f.parent()
Symbolic Ring
sage: [u.parent() for u in [x,y,z]]
[Symbolic Ring, Symbolic Ring, Symbolic Ring]

This means that f is a symbolic function, whose arguments are symbolic variables. Without further assumptions, Sage will treat the value of f as an expression involving members of the symbolic ring, which is roughly equivalent to treat them as complexes. This implies that multiplication is commutative, hence your result.

Frederic's solution will indeed work, because Sage will not attempt to simplify your code for you, and won't simplify x*y-y*x as 0.

Emmanuel Charpentier ( 6 years ago )

Thanks for the clarification!

Thrash ( 6 years ago )

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Nice ! Didn't think of that...

It turns out that you can even use a "function-call" syntax. With a slight change of notations :

sage: R.<a,b,c> = FreeAlgebra(QQbar, 3)
sage: f=(a*b-b*a)*c
sage: f
a*b*c - b*a*c
sage: %cpaste
Pasting code; enter '--' alone on the line to stop or use Ctrl-D.
:A = Matrix(QQ, [[0,1],[0,0]])
:B = Matrix(QQ, [[1,1],[1,1]])
:C = Matrix(QQ, [[0,-1],[-1,0]])
:--
sage: f(A,B,C)
[ 0 -1]
[ 1  0]

which is equivalent (and faster to type) to :

sage: f.subs({a:A, b:B, c:C})
[ 0 -1]
[ 1  0]

Emmanuel Charpentier ( 6 years ago )

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Evaluate polynomial expression on matrices

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Evaluate polynomial expression on matrices