Hello,
Computing the inverse of the identity matrix is not possible. Ok, it
works for rings using RingElement class for the elements (like ZZ, QQ,
RR, CC, ...).
sage: SF = SymmetricFunctions(QQ).schur(); SF
Symmetric Functions over Rational Field in the Schur basis
sage: one = SF.one()
sage: zero = SF.zero()
sage: M = Matrix([[one, zero], [zero, one]])
sage: M
[s[] 0]
[ 0 s[]]
sage: M.det()
s[]
sage: M.is_invertible()
True
sage: M.inverse()
...
AttributeError: 'SymmetricFunctionAlgebra_schur_with_category' object
has no attribute 'fraction_field'
sage: M = Matrix([[one]])
...
AttributeError: 'tuple' object has no attribute 'parent'
sage: M*M
[s[] 0]
[ 0 s[]]
sage: SteenrodAlgebra(7)
mod 7 Steenrod algebra, milnor basis
sage: A = SteenrodAlgebra(7)
sage: M = Matrix([[A.one(), A.zero()], [A.zero(), A.one()]])
sage: M^2
[1 0]
[0 1]
sage: M.inverse()
AttributeError: 'SteenrodAlgebra_generic_with_category' object has no
attribute 'fraction_field'
For the curious, defining a fraction_field for these rings is not
enought. The good fix should be more serious than that.
More generally, there is no any linear algebra in Sage for general
rings. As I worked (the last 6 months) around the coinvariants of the
symmetric group (with a module of rank n! over the symmetric polynomials
in n variables), I only work locally on my machine with a hard hack that
breaks linear algebra over classical rings. I REALLY do not know how to
fix that... ( see http://trac.sagemath.org/ticket/15160 for curious
people, the fix proposed in attachment breaks scalar multiplication for
vector whose element use RingElement class, so it breaks Sage
horribly... ).
Sage is able to model impressively my ring as an abstract free module
over the symmetric functions with 5 different bases (Parent with
multiples realizations : Harmonic polynomials, Schubert polynomials,
Descents monomials, monomials under the staircase and higher Specht
polynomials are the available bases). I have most of basis changes
implemented on my machine but I do need my hack for inverse the
coercions between these bases (since inverting a matrix with unital
determinant over general ring need a fix).
I am ok with the fact that such a bug concerns only people working with
exotic rings or people needing to defined their own rings (like in
algebraic combinatorics, we do need vectors spaces indexed by
anything....). My problems can't be easily translated as problem of
linear algebra over QQ.
Cheers,
Nicolas B.
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