Matrices over Cyclotomic Fields#
The underlying matrix for a Matrix_cyclo_dense object is stored as follows: given an n x m matrix over a cyclotomic field of degree d, we store a d x (nm) matrix over QQ, each column of which corresponds to an element of the original matrix. This can be retrieved via the _rational_matrix method. Here is an example illustrating this:
EXAMPLES:
sage: F.<zeta> = CyclotomicField(5)
sage: M = Matrix(F, 2, 3, [zeta, 3, zeta**4+5, (zeta+1)**4, 0, 1])
sage: M
[ zeta 3 -zeta^3 - zeta^2 - zeta + 4]
[3*zeta^3 + 5*zeta^2 + 3*zeta 0 1]
sage: M._rational_matrix()
[ 0 3 4 0 0 1]
[ 1 0 -1 3 0 0]
[ 0 0 -1 5 0 0]
[ 0 0 -1 3 0 0]
- AUTHORS:
William Stein
Craig Citro
- class sage.matrix.matrix_cyclo_dense.Matrix_cyclo_dense#
Bases:
sage.matrix.matrix_dense.Matrix_dense
Initialize a newly created cyclotomic matrix.
INPUT:
parent
– a matrix space over a cyclotomic number fieldentries
– seematrix()
copy
– ignored (for backwards compatibility)coerce
– if False, assume without checking that the entries lie in the base ring
EXAMPLES:
This function is called implicitly when you create new cyclotomic dense matrices:
sage: W.<a> = CyclotomicField(100) sage: A = matrix(2, 3, [1, 1/a, 1-a,a, -2/3*a, a^19]) sage: A [ 1 -a^39 + a^29 - a^19 + a^9 -a + 1] [ a -2/3*a a^19] sage: TestSuite(A).run()
- charpoly(var='x', algorithm='multimodular', proof=None)#
Return the characteristic polynomial of self, as a polynomial over the base ring.
INPUT:
algorithm
‘multimodular’ (default): reduce modulo primes, compute charpoly mod p, and lift (very fast)
‘pari’: use pari (quite slow; comparable to Magma v2.14 though)
‘hessenberg’: put matrix in Hessenberg form (double dog slow)
proof – bool (default: None) proof flag determined by global linalg proof.
OUTPUT:
polynomial
EXAMPLES:
sage: K.<z> = CyclotomicField(5) sage: a = matrix(K, 3, [1,z,1+z^2, z/3,1,2,3,z^2,1-z]) sage: f = a.charpoly(); f x^3 + (z - 3)*x^2 + (-16/3*z^2 - 2*z)*x - 2/3*z^3 + 16/3*z^2 - 5*z + 5/3 sage: f(a) [0 0 0] [0 0 0] [0 0 0] sage: a.charpoly(algorithm='pari') x^3 + (z - 3)*x^2 + (-16/3*z^2 - 2*z)*x - 2/3*z^3 + 16/3*z^2 - 5*z + 5/3 sage: a.charpoly(algorithm='hessenberg') x^3 + (z - 3)*x^2 + (-16/3*z^2 - 2*z)*x - 2/3*z^3 + 16/3*z^2 - 5*z + 5/3 sage: Matrix(K, 1, [0]).charpoly() x sage: Matrix(K, 1, [5]).charpoly(var='y') y - 5 sage: Matrix(CyclotomicField(13),3).charpoly() x^3 sage: Matrix(CyclotomicField(13),3).charpoly()[2].parent() Cyclotomic Field of order 13 and degree 12
- coefficient_bound()#
Return an upper bound for the (complex) absolute values of all entries of self with respect to all embeddings.
Use
self.height()
for a sharper bound.This is computed using just the Cauchy-Schwarz inequality, i.e., we use the fact that
\left| \sum_i a_i\zeta^i \right| \leq \sum_i |a_i|,
as \(|\zeta| = 1\).
EXAMPLES:
sage: W.<z> = CyclotomicField(5) sage: A = matrix(W, 2, 2, [1+z, 0, 9*z+7, -3 + 4*z]); A [ z + 1 0] [9*z + 7 4*z - 3] sage: A.coefficient_bound() 16
The above bound is just \(9 + 7\), coming from the lower left entry. A better bound would be the following:
sage: (A[1,0]).abs() 12.997543663...
- denominator()#
Return the denominator of the entries of this matrix.
- OUTPUT:
- integer – the smallest integer d so that d * self has
entries in the ring of integers
EXAMPLES:
sage: W.<z> = CyclotomicField(5) sage: A = matrix(W, 2, 2, [-2/7,2/3*z+z^2,-z,1+z/19]); A [ -2/7 z^2 + 2/3*z] [ -z 1/19*z + 1] sage: d = A.denominator(); d 399
- echelon_form(algorithm='multimodular', height_guess=None)#
Find the echelon form of self, using the specified algorithm.
The result is cached for each algorithm separately.
EXAMPLES:
sage: W.<z> = CyclotomicField(3) sage: A = matrix(W, 2, 3, [1+z, 2/3, 9*z+7, -3 + 4*z, z, -7*z]); A [ z + 1 2/3 9*z + 7] [4*z - 3 z -7*z] sage: A.echelon_form() [ 1 0 -192/97*z - 361/97] [ 0 1 1851/97*z + 1272/97] sage: A.echelon_form(algorithm='classical') [ 1 0 -192/97*z - 361/97] [ 0 1 1851/97*z + 1272/97]
We verify that the result is cached and that the caches are separate:
sage: A.echelon_form() is A.echelon_form() True sage: A.echelon_form() is A.echelon_form(algorithm='classical') False
- height()#
Return the height of self.
If we let \(a_{ij}\) be the \(i,j\) entry of self, then we define the height of self to be
\(\max_v \max_{i,j} |a_{ij}|_v\),
where \(v\) runs over all complex embeddings of
self.base_ring()
.EXAMPLES:
sage: W.<z> = CyclotomicField(5) sage: A = matrix(W, 2, 2, [1+z, 0, 9*z+7, -3 + 4*z]); A [ z + 1 0] [9*z + 7 4*z - 3] sage: A.height() 12.997543663... sage: (A[1,0]).abs() 12.997543663...
- randomize(density=1, num_bound=2, den_bound=2, distribution=None, nonzero=False, *args, **kwds)#
Randomize the entries of
self
.Choose rational numbers according to
distribution
, whose numerators are bounded bynum_bound
and whose denominators are bounded byden_bound
.EXAMPLES:
sage: A = Matrix(CyclotomicField(5),2,2,range(4)) ; A [0 1] [2 3] sage: A.randomize() sage: A # random output [ 1/2*zeta5^2 + zeta5 1/2] [ -zeta5^2 + 2*zeta5 -2*zeta5^3 + 2*zeta5^2 + 2]
- set_immutable()#
Change this matrix so that it is immutable.
EXAMPLES:
sage: W.<z> = CyclotomicField(5) sage: A = matrix(W, 2, 2, [1,2/3*z+z^2,-z,1+z/2]) sage: A[0,0] = 10 sage: A.set_immutable() sage: A[0,0] = 20 Traceback (most recent call last): ... ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
Note that there is no function to set a matrix to be mutable again, since such a function would violate the whole point. Instead make a copy, which is always mutable by default.:
sage: A.set_mutable() Traceback (most recent call last): ... AttributeError: 'sage.matrix.matrix_cyclo_dense.Matrix_cyclo_dense' object has no attribute 'set_mutable' sage: B = A.__copy__() sage: B[0,0] = 20 sage: B[0,0] 20
- tensor_product(A, subdivide=True)#
Return the tensor product of two matrices.
INPUT:
A
– a matrixsubdivide
– (default:True
) whether or not to return natural subdivisions with the matrix
OUTPUT:
Replace each element of
self
by a copy ofA
, but first create a scalar multiple ofA
by the element it replaces. So ifself
is an \(m\times n\) matrix andA
is a \(p\times q\) matrix, then the tensor product is an \(mp\times nq\) matrix. By default, the matrix will be subdivided into submatrices of size \(p\times q\).EXAMPLES:
sage: C = CyclotomicField(12) sage: M = matrix.random(C, 3, 3) sage: N = matrix.random(C, 50, 50) sage: M.tensor_product(M) == super(type(M), M).tensor_product(M) True sage: N = matrix.random(C, 15, 20) sage: M.tensor_product(N) == super(type(M), M).tensor_product(N) True