From f69efec4cf990b70763ed18a1cb712fff0a8522d Mon Sep 17 00:00:00 2001
From: Darij Grinberg <darijgrinberg@gmail.com>
Date: Thu, 13 Mar 2025 22:04:10 -0400
Subject: [PATCH] define minpoly_lin for matrices over any field
---
src/sage/matrix/matrix2.pyx | 55 +++++++++++++++++++++++++++++++++++++
1 file changed, 55 insertions(+)
diff --git a/src/sage/matrix/matrix2.pyx b/src/sage/matrix/matrix2.pyx
index 852d749e1e6..e73587d4f75 100644
--- a/src/sage/matrix/matrix2.pyx
+++ b/src/sage/matrix/matrix2.pyx
@@ -3035,6 +3035,61 @@ cdef class Matrix(Matrix1):
self.cache('minpoly', mp)
return mp
+ def minpoly_lin(self, var='x', **kwds):
+ r"""
+ Return the minimal polynomial of ``self``.
+
+ This uses a purely linear-algebraic algorithm (essentially
+ Gaussian elimination, applied to the powers of ``self``).
+ It is slow but it requires no factorization of polynomials.
+
+ EXAMPLES::
+
+ sage: # needs sage.rings.finite_rings
+ sage: A = matrix(GF(9, 'c'), 4, [1,1,0,0, 0,1,0,0, 0,0,5,0, 0,0,0,5])
+ sage: A.minpoly_lin()
+ x^3 + 2*x^2 + 2*x + 1
+ sage: A.minpoly_lin()(A) == 0
+ True
+ sage: CF = CyclotomicField()
+ sage: i = CF.gen(4)
+ sage: A = matrix(CF, 4, [1,1,0,0, 0,1,0,0, 0,0,1+i,0, 0,0,0,-i])
+ sage: A.minpoly_lin()
+ x^4 - 3*x^3 + (4 - E(4))*x^2 + (-3 + 2*E(4))*x + 1 - E(4)
+
+ The default variable name is `x`, but you can specify
+ another name::
+
+ sage: # needs sage.rings.finite_rings
+ sage: A.minpoly_lin('y')
+ y^4 - 3*y^3 + (4 - E(4))*y^2 + (-3 + 2*E(4))*y + 1 - E(4)
+ """
+ f = self.fetch('minpoly')
+ if f is not None:
+ return f.change_variable_name(var)
+
+ n = self.nrows()
+ F = self.base_ring()
+ from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+ P = PolynomialRing(F, "x")
+ x = P.gen()
+ from sage.modules.free_module_element import vector
+ from sage.matrix.constructor import matrix
+
+ pow = self**0
+ pows = []
+ for k in range(n+1):
+ pows.append(vector(pow.list()))
+ pow *= self
+ Mpows = matrix(pows)
+ try:
+ cs = Mpows.solve_left(vector(pow.list()))
+ except ValueError:
+ continue
+ mp = x**(k+1) - P.sum(cs[i] * x**i for i in range(k+1))
+ self.cache('minpoly', mp)
+ return mp
+
def _test_minpoly(self, **options):
"""
Check that :meth:`minpoly` works.