Re: [Axiom-developer] evaluate a polynom for a matrix.

Please direct questions of this kind to [email protected].

Francois Maltey <[email protected]> writes:

> Hello,
> 
> Let P in Ring[X] and M in SquareMatrix(Ring)
> 
> The mathematics evaluate P(M) 
>    by sum (ak*M^k, k=0..deg P) in SquareMatrix(Ring)
> where P = sum (ak*X^k, k=0..deg P)
> 
> How can I do this in axiom.

-- work in SquareMatrix(n, FRAC POLY INT)

n := 4;

SM ==> SquareMatrix(n, FRAC POLY INT)

--
-- Create a generic matrix
--

M: SM := matrix [[a[i,j] for j in 1..n] for i in 1..n]

-- Create an identity matrix

I: SM := 1;

-- Computes determinant (M - x*Id) where M is the previous generic matrix.
-- note that characteristicPolynomial would do this for youm however, for some
-- strange reason, it accepts an element of Matrix R instead of SquareMatrix R.

P := determinant (M - x * I)

-- Now evaluate P(M). We find the (0)_4 matrix.  Note that you can coerce only
-- to a univariate polynomial, since matrix multiplication is not
-- commutative...

(P::UP(x, SM))(M)