On 05/13/2006 10:35 PM, Fabio S. wrote:
BTW, I also have another question on this topic: if I compute the
characteristic polynomial of a matrix and then I evaluate the
polynomial at the matrix, I would expect to have the zero matrix as
result. This does not happens: try
m:=matrix([[1,2],[3,4]])
p:=characteristicPolynomial(m)
p(m)
Why?
Because Axiom is not smart enough to figure out all the types for you.
Does this make you happier?
M := SquareMatrix(2,Integer)
m:M:=matrix([[1,2],[3,4]])
P:=SUP(M)
p: P := characteristicPolynomial(m)::P
x := first variables p
eval(p, x, m)
I must say that this doesn't make me happier at all. I didn't expect
that evaluating a polynomial at a matrix was such a terrible task for
the type resolver in axiom: the result of characteristicPolynomial(m) is
a polynomial with integer coefficient in one variable. I can't see the
difficulty to convert it to UP SquareMatrix(2,Integer) (of course, just
from a mathematical point of view).
Well, as I said SUP(Integer) should have been enough. But now, what you
actually want is NOT a polynomial. You want a *function*
charpol: SquareMatrix(2, Integer) -> SquareMatrix(2, Integer)
and that is something different. I am sure you know the difference.
The big trouble now is to convert the polynomial you get into a
polynomial function with the above type. I guess the interpreter would
have been able to find such a conversion if there were an implementation
of an evaluation of a polynomial at a *square* matrix (arbitrary
matrices would, of course, not work). However, my browsing through
hyperdoc was unsuccessful. So I guess the non-existence of such an
implementation makes the interpreter telling you about the problem.
Moreover, given the Hamilton-Cayley, it doesn't seem to me a very
bizzarre idea to try to evaluate the char pol of a matrix at the matrix
itself...
No, of course, not. But as far as I can tell, that is not directly
implemented.
I would expect that this is the first example for every linear algebra
student, so I would expect that the implementation of the function
should take care of this possibility.
I somehow have the impression that is one of the reasons why Axiom
failed commercially. Simple things must be expressed in a complicated
manner. I agree that Axiom is not (yet) perfect. Just help and provide a
implementation for such a function and make all the people who come
after you happier.
Thank you
Ralf
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