"""nodoctest
Matrices over a domain
"""
cimport matrix
import matrix
import sage.structure.sequence
cdef class Matrix_domain_dense(matrix.Matrix):
def charpoly(self, *args, **kwds):
r"""
Return the characteristic polynomial of self, as a polynomial
over the base ring.
ALGORITHM: Use \code{self.matrix_over_field()} to obtain the
corresponding matrix over a field, then call the
\code{charpoly} function on it.
EXAMPLES:
First a matrix over $\Z$:
sage: A = MatrixSpace(IntegerRing(),2)( [[1,2], [3,4]] )
sage: f = A.charpoly('x')
sage: f
x^2 - 5*x - 2
sage: f.parent()
Univariate Polynomial Ring in x over Integer Ring
We compute the characteristic polynomial of a matrix over
the polynomial ring $\Z[a]$:
sage: R = PolynomialRing(IntegerRing(),'a'); a = R.gen()
sage: M = MatrixSpace(R,2)([[a,1], [a,a+1]])
sage: M
[ a 1]
[ a a + 1]
sage: f = M.charpoly('x')
sage: f
x^2 + (-2*a - 1)*x + a^2
sage: f.parent()
Univariate Polynomial Ring in x over Univariate Polynomial Ring in a over Integer Ring
sage: M.trace()
2*a + 1
sage: M.determinant()
a^2
We compute the characteristic polynomial of a matrix over the
multi-variate polynomial ring $\Z[x,y]$:
sage: R = PolynomialRing(IntegerRing(),2); x,y = R.gens()
sage: A = MatrixSpace(R,2)([x, y, x^2, y^2])
sage: f = A.charpoly('x')
sage: f
x^2 + (-1*x1^2 - x0)*x + x0*x1^2 - x0^2*x1
It's a little difficult to distinguish the variables. To fix this,
we rename the indeterminate $Z$:
sage: f.parent()._assign_names("Z")
sage: f
Z^2 + (-1*x1^2 - x0)*Z + x0*x1^2 - x0^2*x1
We can pass parameters in, which are passed on to the charpoly
function for matrices over a field.
sage: A = 1000*MatrixSpace(ZZ,10)(range(100))
sage: A.charpoly(bound=2)
x^10 + 14707*x^9 - 21509*x^8
sage: A = 1000*MatrixSpace(ZZ,10)(range(100))
sage: A.charpoly('x')
x^10 - 495000*x^9 - 8250000000*x^8
"""
f = self.matrix_over_field(copy=True).charpoly(*args, **kwds)
return f.base_extend(self.base_ring())
def determinant(self):
"""
Return the determinant of this matrix.
INPUT:
-- a square matrix
ALGORITHM: Find the characteristic polynomial and take its
constant term (up to sign).
EXAMPLES:
We create a matrix over $\Z[x,y]$ and compute its determinant.
sage: R = PolynomialRing(IntegerRing(),2); x,y = R.gens()
sage: A = MatrixSpace(R,2)([x, y, x**2, y**2])
sage: A.determinant()
x0*x1^2 - x0^2*x1
"""
if self.nrows() != self.ncols():
raise ArithmeticError("Matrix must be square, but is %sx%s"%(
self.nrows(), self.ncols()))
d = (-1)**self.nrows() * self.charpoly('x')[0]
return self.base_ring()(d)
def is_invertible(self):
r"""
Return True if this matrix is invertible.
EXAMPLES:
The following matrix is invertible over $\Q$ but not over $\Z$.
sage: A = MatrixSpace(IntegerRing(), 2)(range(4))
sage: A.is_invertible()
False
sage: A.matrix_over_field().is_invertible()
True
The inverse function is a constructor for matrices over the
fraction field, so it can work even if A is not invertible.
sage: ~A # inverse of A
[-3/2 1/2]
[ 1 0]
The next matrix is invertible over $\Z$.
sage: A = MatrixSpace(IntegerRing(),2)([1,10,0,-1])
sage: A.is_invertible()
True
sage: ~A # compute the inverse
[ 1 10]
[ 0 -1]
The following nontrivial matrix is invertible over $\Z[x]$.
sage: R = PolynomialRing(IntegerRing())
sage: x = R.gen()
sage: A = MatrixSpace(R,2)([1,x,0,-1])
sage: A.is_invertible()
True
sage: ~A
[ 1 x]
[ 0 -1]
"""
return self.is_square() and self.determinant().is_unit()
def __invert__(self):
r"""
Return this inverse of this matrix, as a matrix over the fraction field.
Raises a \code{ZeroDivisionError} if the matrix has zero
determinant, and raises an \code{ArithmeticError}, if the
inverse doesn't exist because the matrix is nonsquare.
EXAMPLES:
sage: A = MatrixSpace(IntegerRing(), 2)([1,1,3,5])
sage: ~A
[ 5/2 -1/2]
[-3/2 1/2]
Even if the inverse lies in the base field, the result is still a matrix
over the fraction field.
sage: I = MatrixSpace(IntegerRing(),2)( 1 ) # identity matrix
sage: ~I
[1 0]
[0 1]
sage: (~I).parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
TESTS:
sage: MatrixSpace(IntegerRing(), 0)().inverse()
[]
"""
return ~self.matrix_over_field()
def matrix_over_field(self, copy=False):
"""
Return this matrix, but with entries viewed as elements
of the fraction field of the base ring.
EXAMPLES:
sage: A = MatrixSpace(IntegerRing(),2)([1,2,3,4])
sage: B = A.matrix_over_field()
sage: B
[1 2]
[3 4]
sage: B.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
"""
return self.change_ring(self.base_ring().fraction_field(), copy=copy)
def _singular_(self, singular=None):
"""
Tries to coerce this matrix to a singular matrix.
"""
if singular is None:
from sage.interfaces.all import singular as singular_default
singular = singular_default
try:
self.base_ring()._singular_(singular)
except (NotImplementedError, AttributeError):
raise TypeError("Cannot coerce to Singular")
return singular.matrix(self.nrows(),self.ncols(),singular(self.list()))
