I tried to demonstrate Cayley Hamilton in Sage, but failed. Here is what I
tries:
sage: f = function('f')
sage: m = matrix([[f(i,j) for j in range(2)] for i in range(2)])
sage: p=SR[x](m.characteristic_polynomial('x'))
sage: p.subs(x=m)
[(f(0, 0) - x)*(f(1, 1) - x) - f(0, 1)*f(1, 0)
0]
[ 0 (f(0, 0) - x)*(f(1, 1) - x) -
f(0, 1)*f(1, 0)]
Of course, the result *should* be the zero matrix. It seems that the value of
p is not what I'd expect:
sage: p.coefficients()
[(f(0, 0) - x)*(f(1, 1) - x) - f(0, 1)*f(1, 0)]
So, probably the question is: how do I create a polynomial over Symbolic Ring
properly?
Many thanks,
Martin
I'm not sure whether it helps, but here is how I'd do it in FriCAS:
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]
-- Computes determinant (M - x*Id) where M is the previous generic matrix.
P := determinant (M - x * 1)
-- Now evaluate P(M). We find the (0)_4 matrix. Note that you can coerce only
-- to a univariate polynomial (short: UP), since matrix multiplication is not
-- commutative...
(P::UP(x, SM)) M
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