Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
""" Base class for matrices, part 1
For design documentation see :mod:`sage.matrix.docs`.
TESTS::
sage: A = Matrix(GF(5),3,3,srange(9)) sage: TestSuite(A).run() """
#***************************************************************************** # Copyright (C) 2005, 2006 William Stein <wstein@gmail.com> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #***************************************************************************** from __future__ import absolute_import
from cpython cimport *
import sage.modules.free_module from sage.structure.element cimport coercion_model
cdef class Matrix(Matrix0): ################################################### # Coercion to Various Systems ###################################################
def _pari_init_(self): """ Return a string defining a GP representation of self.
EXAMPLES::
sage: R.<x> = QQ['x'] sage: a = matrix(R,2,[x+1,2/3, x^2/2, 1+x^3]); a [ x + 1 2/3] [1/2*x^2 x^3 + 1] sage: b = gp(a); b # indirect doctest [x + 1, 2/3; 1/2*x^2, x^3 + 1] sage: a.determinant() x^4 + x^3 - 1/3*x^2 + x + 1 sage: b.matdet() x^4 + x^3 - 1/3*x^2 + x + 1 """ cdef Py_ssize_t nr, nc, i, j
def __pari__(self): """ Return the Pari matrix corresponding to self.
EXAMPLES::
sage: R.<x> = QQ['x'] sage: a = matrix(R,2,[x+1,2/3, x^2/2, 1+x^3]); a [ x + 1 2/3] [1/2*x^2 x^3 + 1] sage: b = pari(a); b # indirect doctest [x + 1, 2/3; 1/2*x^2, x^3 + 1] sage: a.determinant() x^4 + x^3 - 1/3*x^2 + x + 1 sage: b.matdet() x^4 + x^3 - 1/3*x^2 + x + 1
This function preserves precision for entries of inexact type (e.g. reals)::
sage: R = RealField(4) # 4 bits of precision sage: a = matrix(R, 2, [1, 2, 3, 1]); a [1.0 2.0] [3.0 1.0] sage: b = pari(a); b [1.000000000, 2.000000000; 3.000000000, 1.000000000] # 32-bit [1.00000000000000, 2.00000000000000; 3.00000000000000, 1.00000000000000] # 64-bit sage: b[0][0].precision() # in words 3 """
def _gap_init_(self): """ Returns a string defining a gap representation of self.
EXAMPLES::
sage: A = MatrixSpace(QQ,3,3)([0,1,2,3,4,5,6,7,8]) sage: g = gap(A) # indirect doctest sage: g [ [ 0, 1, 2 ], [ 3, 4, 5 ], [ 6, 7, 8 ] ] sage: g.CharacteristicPolynomial() x_1^3-12*x_1^2-18*x_1 sage: A.characteristic_polynomial() x^3 - 12*x^2 - 18*x sage: matrix(QQ, g) == A True
Particularly difficult is the case of matrices over cyclotomic fields and general number fields. See :trac:`5618` and :trac:`8909`::
sage: K.<zeta> = CyclotomicField(8) sage: A = MatrixSpace(K,2,2)([0,1+zeta,2*zeta,3]) sage: g = gap(A); g [ [ 0, 1+E(8) ], [ 2*E(8), 3 ] ] sage: matrix(K, g) == A True sage: g.IsMatrix() true
sage: L.<tau> = NumberField(x^3-2) sage: A = MatrixSpace(L,2,2)([0,1+tau,2*tau,3]) sage: g = gap(A); g [ [ !0, tau+1 ], [ 2*tau, !3 ] ] sage: matrix(L, g) == A True """ cdef Py_ssize_t i, j # It is needed to multiply with 'One(...)', because # otherwise the result would not be a gap matrix
def _libgap_(self): """ Construct a LibGAP matrix.
INPUT:
- ``M`` -- a matrix.
OUTPUT:
A GAP matrix, that is, a list of lists with entries over a common ring.
EXAMPLES::
sage: libgap(identity_matrix(ZZ,2)) [ [ 1, 0 ], [ 0, 1 ] ] sage: libgap(matrix(GF(3),2,2,[4,5,6,7])) [ [ Z(3)^0, Z(3) ], [ 0*Z(3), Z(3)^0 ] ] """
def _giac_init_(self): """ Return a Giac string representation of this matrix.
EXAMPLES::
sage: M = matrix(ZZ,2,range(4)) sage: giac(M) [[0,1],[2,3]]
::
sage: M = matrix(QQ,3,[1,2,3,4/3,5/3,6/4,7,8,9]) sage: giac(M) [[1,2,3],[4/3,5/3,3/2],[7,8,9]]
::
sage: P.<x> = ZZ[] sage: M = matrix(P, 2, [-9*x^2-2*x+2, x-1, x^2+8*x, -3*x^2+5]) sage: giac(M) [[-9*x^2-2*x+2,x-1],[x^2+8*x,-3*x^2+5]] """
def _maxima_init_(self): """ Return a string representation of this matrix in Maxima.
EXAMPLES::
sage: m = matrix(3,range(9)); m [0 1 2] [3 4 5] [6 7 8] sage: m._maxima_init_() 'matrix([0,1,2],[3,4,5],[6,7,8])' sage: a = maxima(m); a matrix([0,1,2],[3,4,5],[6,7,8]) sage: a.charpoly('x').expand() (-x^3)+12*x^2+18*x sage: m.charpoly() x^3 - 12*x^2 - 18*x """ cdef Py_ssize_t i, j
def _mathematica_init_(self): """ Return Mathematica string representation of this matrix.
EXAMPLES::
sage: A = MatrixSpace(QQ,3)([1,2,3,4/3,5/3,6/4,7,8,9]) sage: g = mathematica(A); g # optional - mathematica {{1, 2, 3}, {4/3, 5/3, 3/2}, {7, 8, 9}} sage: A._mathematica_init_() '{{1/1, 2/1, 3/1}, {4/3, 5/3, 3/2}, {7/1, 8/1, 9/1}}'
::
sage: A = matrix([[1,2],[3,4]]) sage: g = mathematica(A); g # optional - mathematica {{1, 2}, {3, 4}}
::
sage: a = matrix([[pi, sin(x)], [cos(x), 1/e]]); a [ pi sin(x)] [cos(x) e^(-1)] sage: a._mathematica_init_() '{{Pi, Sin[x]}, {Cos[x], Exp[-1]}}' """
def _magma_init_(self, magma): r""" Return a string that evaluates in the given Magma session to this matrix.
EXAMPLES:
We first coerce a square matrix.
::
sage: A = MatrixSpace(QQ,3)([1,2,3,4/3,5/3,6/4,7,8,9]) sage: B = magma(A); B # (indirect doctest) optional - magma [ 1 2 3] [4/3 5/3 3/2] [ 7 8 9] sage: B.Type() # optional - magma AlgMatElt sage: B.Parent() # optional - magma Full Matrix Algebra of degree 3 over Rational Field
We coerce a non-square matrix over `\ZZ/8\ZZ`.
::
sage: A = MatrixSpace(Integers(8),2,3)([-1,2,3,4,4,-2]) sage: B = magma(A); B # optional - magma [7 2 3] [4 4 6] sage: B.Type() # optional - magma ModMatRngElt sage: B.Parent() # optional - magma Full RMatrixSpace of 2 by 3 matrices over IntegerRing(8)
::
sage: R.<x,y> = QQ[] sage: A = MatrixSpace(R,2,2)([x+y,x-1,y+5,x*y]) sage: B = magma(A); B # optional - magma [x + y x - 1] [y + 5 x*y]
::
sage: R.<x,y> = ZZ[] sage: A = MatrixSpace(R,2,2)([x+y,x-1,y+5,x*y]) sage: B = magma(A); B # optional - magma [x + y x - 1] [y + 5 x*y]
We coerce a matrix over a cyclotomic field, where the generator must be named during the coercion.
::
sage: K = CyclotomicField(9) ; z = K.0 sage: M = matrix(K,3,3,[0,1,3,z,z**4,z-1,z**17,1,0]) sage: M [ 0 1 3] [ zeta9 zeta9^4 zeta9 - 1] [-zeta9^5 - zeta9^2 1 0] sage: magma(M) # optional - magma [ 0 1 3] [ zeta9 zeta9^4 zeta9 - 1] [-zeta9^5 - zeta9^2 1 0] sage: magma(M**2) == magma(M)**2 # optional - magma True """ P = magma(self.parent()) v = [x._magma_init_(magma) for x in self.list()] return '%s![%s]'%(P.name(), ','.join(v))
def _maple_init_(self): """ Return a Maple string representation of this matrix.
EXAMPLES::
sage: M = matrix(ZZ,2,range(4)) sage: maple(M) # optional - maple Matrix(2, 2, [[0,1],[2,3]])
::
sage: M = matrix(QQ,3,[1,2,3,4/3,5/3,6/4,7,8,9]) sage: maple(M) # optional - maple Matrix(3, 3, [[1,2,3],[4/3,5/3,3/2],[7,8,9]])
::
sage: P.<x> = ZZ[] sage: M = matrix(P, 2, [-9*x^2-2*x+2, x-1, x^2+8*x, -3*x^2+5]) sage: maple(M) # optional - maple Matrix(2, 2, [[-9*x^2-2*x+2,x-1],[x^2+8*x,-3*x^2+5]]) """ s = str(self.rows()).replace('(','[').replace(')',']') return "Matrix(%s,%s,%s)"%(self.nrows(), self.ncols(), s)
def _polymake_(self, polymake=None): """ Tries to coerce this matrix to a polymake matrix.
EXAMPLES::
sage: M = matrix(ZZ,2,range(4)) sage: polymake(M) # optional - polymake 0 1 2 3 sage: K.<sqrt5> = QuadraticField(5) sage: M = matrix(K, [[1, 2], [sqrt5, 3]]) sage: polymake(M) # optional - polymake 1 2 0+1r5 3 """ P = polymake(self.parent()) return polymake.new_object(P, [ list(r) for r in self.rows(copy=False) ])
def _singular_(self, singular=None): """ Tries to coerce this matrix to a singular matrix. """ except (NotImplementedError, AttributeError): raise TypeError("Cannot coerce to Singular")
def _macaulay2_(self, macaulay2=None): """ EXAMPLES::
sage: m = matrix(ZZ, [[1,2],[3,4]]) sage: macaulay2(m) #optional - macaulay2 (indirect doctest) | 1 2 | | 3 4 |
::
sage: R.<x,y> = QQ[] sage: m = matrix([[x,y],[1+x,1+y]]) sage: macaulay2(m) #optional - macaulay2 | x y | | x+1 y+1 | """ base_ring = macaulay2(self.base_ring()) entries = map(list, self) return macaulay2(entries).matrix()
def _scilab_init_(self): """ Returns a string defining a Scilab representation of self.
EXAMPLES:
sage: a = matrix([[1,2,3],[4,5,6],[7,8,9]]); a [1 2 3] [4 5 6] [7 8 9] sage: a._scilab_init_() '[1,2,3;4,5,6;7,8,9]'
AUTHORS:
- Ronan Paixao (2008-12-12) """ cdef Py_ssize_t nr, nc, i, j
def _scilab_(self, scilab=None): """ Creates a ScilabElement object based on self and returns it.
EXAMPLES:
sage: a = matrix([[1,2,3],[4,5,6],[7,8,9]]); a [1 2 3] [4 5 6] [7 8 9] sage: b = scilab(a); b # optional - scilab (indirect doctest) 1. 2. 3. 4. 5. 6. 7. 8. 9.
AUTHORS:
- Ronan Paixao (2008-12-12) """ return scilab(self._scilab_init_())
def _sage_input_(self, sib, coerce): r""" Produce an expression which will reproduce this value when evaluated.
EXAMPLES::
sage: sage_input(matrix(QQ, 3, 3, [5..13])/7, verify=True) # Verified matrix(QQ, [[5/7, 6/7, 1], [8/7, 9/7, 10/7], [11/7, 12/7, 13/7]]) sage: sage_input(MatrixSpace(GF(5), 50, 50, sparse=True).random_element(density=0.002), verify=True) # Verified matrix(GF(5), 50, 50, {(4,44):2, (5,25):1, (26,9):3, (43,24):3, (44,38):4}) sage: from sage.misc.sage_input import SageInputBuilder sage: matrix(RDF, [[3, 1], [4, 1]])._sage_input_(SageInputBuilder(), False) {call: {atomic:matrix}({atomic:RDF}, {list: ({list: ({atomic:3}, {atomic:1})}, {list: ({atomic:4}, {atomic:1})})})} sage: matrix(ZZ, 50, 50, {(9,17):1})._sage_input_(SageInputBuilder(), False) {call: {atomic:matrix}({atomic:ZZ}, {atomic:50}, {atomic:50}, {dict: {{atomic:(9,17)}:{atomic:1}}})}
TESTS::
sage: sage_input(matrix(RR, 0, 3, []), verify=True) # Verified matrix(RR, 0, 3) sage: sage_input(matrix(RR, 3, 0, []), verify=True) # Verified matrix(RR, 3, 0) sage: sage_input(matrix(RR, 0, 0, []), verify=True) # Verified matrix(RR, 0, 0) """ # We hand-format the keys to get rid of the space that would # normally follow the comma else:
def numpy(self, dtype=None): """ Return the Numpy matrix associated to this matrix.
INPUT:
- ``dtype`` - The desired data-type for the array. If not given, then the type will be determined as the minimum type required to hold the objects in the sequence.
EXAMPLES::
sage: a = matrix(3,range(12)) sage: a.numpy() array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) sage: a.numpy('f') array([[ 0., 1., 2., 3.], [ 4., 5., 6., 7.], [ 8., 9., 10., 11.]], dtype=float32) sage: a.numpy('d') array([[ 0., 1., 2., 3.], [ 4., 5., 6., 7.], [ 8., 9., 10., 11.]]) sage: a.numpy('B') array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]], dtype=uint8)
Type ``numpy.typecodes`` for a list of the possible typecodes::
sage: import numpy sage: sorted(numpy.typecodes.items()) [('All', '?bhilqpBHILQPefdgFDGSUVOMm'), ('AllFloat', 'efdgFDG'), ('AllInteger', 'bBhHiIlLqQpP'), ('Character', 'c'), ('Complex', 'FDG'), ('Datetime', 'Mm'), ('Float', 'efdg'), ('Integer', 'bhilqp'), ('UnsignedInteger', 'BHILQP')]
Alternatively, numpy automatically calls this function (via the magic :meth:`__array__` method) to convert Sage matrices to numpy arrays::
sage: import numpy sage: b=numpy.array(a); b array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) sage: b.dtype dtype('int32') # 32-bit dtype('int64') # 64-bit sage: b.shape (3, 4) """
# Define the magic "__array__" function so that numpy.array(m) can convert # a matrix m to a numpy array. # See http://docs.scipy.org/doc/numpy/user/c-info.how-to-extend.html#converting-an-arbitrary-sequence-object __array__=numpy
################################################### # Construction functions ###################################################
def matrix_over_field(self): """ Return copy of this matrix, but with entries viewed as elements of the fraction field of the base ring (assuming it is defined).
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 """
def lift(self): """ Return lift of self to the covering ring of the base ring R, which is by definition the ring returned by calling cover_ring() on R, or just R itself if the cover_ring method is not defined.
EXAMPLES::
sage: M = Matrix(Integers(7), 2, 2, [5, 9, 13, 15]) ; M [5 2] [6 1] sage: M.lift() [5 2] [6 1] sage: parent(M.lift()) Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
The field QQ doesn't have a cover_ring method::
sage: hasattr(QQ, 'cover_ring') False
So lifting a matrix over QQ gives back the same exact matrix.
::
sage: B = matrix(QQ, 2, [1..4]) sage: B.lift() [1 2] [3 4] sage: B.lift() is B True """
def lift_centered(self): """ Apply the lift_centered method to every entry of self.
OUTPUT:
If self is a matrix over the Integers mod `n`, this method returns the unique matrix `m` such that `m` is congruent to self mod `n` and for every entry `m[i,j]` we have `-n/2 < m[i,j] \\leq n/2`. If the coefficient ring does not have a cover_ring method, return self.
EXAMPLES::
sage: M = Matrix(Integers(8), 2, 4, range(8)) ; M [0 1 2 3] [4 5 6 7] sage: L = M.lift_centered(); L [ 0 1 2 3] [ 4 -3 -2 -1] sage: parent(L) Full MatrixSpace of 2 by 4 dense matrices over Integer Ring
The returned matrix is congruent to M modulo 8.::
sage: L.mod(8) [0 1 2 3] [4 5 6 7]
The field QQ doesn't have a cover_ring method::
sage: hasattr(QQ, 'cover_ring') False
So lifting a matrix over QQ gives back the same exact matrix.
::
sage: B = matrix(QQ, 2, [1..4]) sage: B.lift_centered() [1 2] [3 4] sage: B.lift_centered() is B True """ pass
############################################################################################# # rows, columns, sparse_rows, sparse_columns, dense_rows, dense_columns, row, column #############################################################################################
def columns(self, copy=True): r""" Return a list of the columns of self.
INPUT:
- ``copy`` - (default: True) if True, return a copy of the list of columns which is safe to change.
If ``self`` is a sparse matrix, columns are returned as sparse vectors, otherwise returned vectors are dense.
EXAMPLES::
sage: matrix(3, [1..9]).columns() [(1, 4, 7), (2, 5, 8), (3, 6, 9)] sage: matrix(RR, 2, [sqrt(2), pi, exp(1), 0]).columns() [(1.41421356237310, 2.71828182845905), (3.14159265358979, 0.000000000000000)] sage: matrix(RR, 0, 2, []).columns() [(), ()] sage: matrix(RR, 2, 0, []).columns() [] sage: m = matrix(RR, 3, 3, {(1,2): pi, (2, 2): -1, (0,1): sqrt(2)}) sage: parent(m.columns()[0]) Sparse vector space of dimension 3 over Real Field with 53 bits of precision
Sparse matrices produce sparse columns. ::
sage: A = matrix(QQ, 2, range(4), sparse=True) sage: v = A.columns()[0] sage: v.is_sparse() True
TESTS::
sage: A = matrix(QQ, 4, range(16)) sage: A.columns('junk') Traceback (most recent call last): ... ValueError: 'copy' must be True or False, not junk """ else:
def rows(self, copy=True): r""" Return a list of the rows of self.
INPUT:
- ``copy`` - (default: True) if True, return a copy of the list of rows which is safe to change.
If ``self`` is a sparse matrix, rows are returned as sparse vectors, otherwise returned vectors are dense.
EXAMPLES::
sage: matrix(3, [1..9]).rows() [(1, 2, 3), (4, 5, 6), (7, 8, 9)] sage: matrix(RR, 2, [sqrt(2), pi, exp(1), 0]).rows() [(1.41421356237310, 3.14159265358979), (2.71828182845905, 0.000000000000000)] sage: matrix(RR, 0, 2, []).rows() [] sage: matrix(RR, 2, 0, []).rows() [(), ()] sage: m = matrix(RR, 3, 3, {(1,2): pi, (2, 2): -1, (0,1): sqrt(2)}) sage: parent(m.rows()[0]) Sparse vector space of dimension 3 over Real Field with 53 bits of precision
Sparse matrices produce sparse rows. ::
sage: A = matrix(QQ, 2, range(4), sparse=True) sage: v = A.rows()[0] sage: v.is_sparse() True
TESTS::
sage: A = matrix(QQ, 4, range(16)) sage: A.rows('junk') Traceback (most recent call last): ... ValueError: 'copy' must be True or False, not junk """ else:
def dense_columns(self, copy=True): """ Return list of the dense columns of self.
INPUT:
- ``copy`` - (default: True) if True, return a copy so you can modify it safely
EXAMPLES:
An example over the integers::
sage: a = matrix(3,3,range(9)); a [0 1 2] [3 4 5] [6 7 8] sage: a.dense_columns() [(0, 3, 6), (1, 4, 7), (2, 5, 8)]
We do an example over a polynomial ring::
sage: R.<x> = QQ[ ] sage: a = matrix(R, 2, [x,x^2, 2/3*x,1+x^5]); a [ x x^2] [ 2/3*x x^5 + 1] sage: a.dense_columns() [(x, 2/3*x), (x^2, x^5 + 1)] sage: a = matrix(R, 2, [x,x^2, 2/3*x,1+x^5], sparse=True) sage: c = a.dense_columns(); c [(x, 2/3*x), (x^2, x^5 + 1)] sage: parent(c[1]) Ambient free module of rank 2 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
TESTS:
Check that the returned rows are immutable as per :trac:`14874`::
sage: m = Mat(ZZ,3,3)(range(9)) sage: v = m.dense_columns() sage: [x.is_mutable() for x in v] [False, False, False] """ if copy: return list(x) return x cdef Py_ssize_t i
# Make the vectors immutable since we are caching them
# cache result else:
def dense_rows(self, copy=True): """ Return list of the dense rows of self.
INPUT:
- ``copy`` - (default: True) if True, return a copy so you can modify it safely (note that the individual vectors in the copy should not be modified since they are mutable!)
EXAMPLES::
sage: m = matrix(3, range(9)); m [0 1 2] [3 4 5] [6 7 8] sage: v = m.dense_rows(); v [(0, 1, 2), (3, 4, 5), (6, 7, 8)] sage: v is m.dense_rows() False sage: m.dense_rows(copy=False) is m.dense_rows(copy=False) True sage: m[0,0] = 10 sage: m.dense_rows() [(10, 1, 2), (3, 4, 5), (6, 7, 8)]
TESTS:
Check that the returned rows are immutable as per :trac:`14874`::
sage: m = Mat(ZZ,3,3)(range(9)) sage: v = m.dense_rows() sage: [x.is_mutable() for x in v] [False, False, False] """
cdef Py_ssize_t i
# Make the vectors immutable since we are caching them
# cache result else:
def sparse_columns(self, copy=True): r""" Return a list of the columns of ``self`` as sparse vectors (or free module elements).
INPUT:
- ``copy`` - (default: True) if True, return a copy so you can modify it safely
EXAMPLES::
sage: a = matrix(2,3,range(6)); a [0 1 2] [3 4 5] sage: v = a.sparse_columns(); v [(0, 3), (1, 4), (2, 5)] sage: v[1].is_sparse() True
TESTS:
Columns of sparse matrices having no columns were fixed on :trac:`10714`::
sage: m = matrix(10, 0, sparse=True) sage: m.ncols() 0 sage: m.columns() []
Check that the returned columns are immutable as per :trac:`14874`::
sage: m = Mat(ZZ,3,3,sparse=True)(range(9)) sage: v = m.sparse_columns() sage: [x.is_mutable() for x in v] [False, False, False] """ if copy: return list(x) return x
cdef Py_ssize_t i, j
# new column -- emit vector C.append(F(0))
# finish up C.append(F(0))
# Make the vectors immutable since we are caching them
# cache and return result else: return C
def sparse_rows(self, copy=True): r""" Return a list of the rows of ``self`` as sparse vectors (or free module elements).
INPUT:
- ``copy`` - (default: True) if True, return a copy so you can modify it safely
EXAMPLES::
sage: m = Mat(ZZ,3,3,sparse=True)(range(9)); m [0 1 2] [3 4 5] [6 7 8] sage: v = m.sparse_rows(); v [(0, 1, 2), (3, 4, 5), (6, 7, 8)] sage: m.sparse_rows(copy=False) is m.sparse_rows(copy=False) True sage: v[1].is_sparse() True sage: m[0,0] = 10 sage: m.sparse_rows() [(10, 1, 2), (3, 4, 5), (6, 7, 8)]
TESTS:
Rows of sparse matrices having no rows were fixed on :trac:`10714`::
sage: m = matrix(0, 10, sparse=True) sage: m.nrows() 0 sage: m.rows() []
Check that the returned rows are immutable as per :trac:`14874`::
sage: m = Mat(ZZ,3,3,sparse=True)(range(9)) sage: v = m.sparse_rows() sage: [x.is_mutable() for x in v] [False, False, False] """
cdef Py_ssize_t i, j
# new row -- emit vector
# finish up
# Make the vectors immutable since we are caching them
# cache and return result else: return R
def column(self, Py_ssize_t i, from_list=False): """ Return the ``i``'th column of this matrix as a vector.
This column is a dense vector if and only if the matrix is a dense matrix.
INPUT:
- ``i`` - integer
- ``from_list`` - bool (default: False); if true, returns the ``i``'th element of ``self.columns()`` (see :func:`columns()`), which may be faster, but requires building a list of all columns the first time it is called after an entry of the matrix is changed.
EXAMPLES::
sage: a = matrix(2,3,range(6)); a [0 1 2] [3 4 5] sage: a.column(1) (1, 4)
If the column is negative, it wraps around, just like with list indexing, e.g., -1 gives the right-most column::
sage: a.column(-1) (2, 5)
TESTS::
sage: a = matrix(2,3,range(6)); a [0 1 2] [3 4 5] sage: a.column(3) Traceback (most recent call last): ... IndexError: column index out of range sage: a.column(-4) Traceback (most recent call last): ... IndexError: column index out of range """ raise IndexError("matrix has no columns") cdef Py_ssize_t j
def row(self, Py_ssize_t i, from_list=False): """ Return the ``i``'th row of this matrix as a vector.
This row is a dense vector if and only if the matrix is a dense matrix.
INPUT:
- ``i`` - integer
- ``from_list`` - bool (default: False); if true, returns the ``i``'th element of ``self.rows()`` (see :func:`rows`), which may be faster, but requires building a list of all rows the first time it is called after an entry of the matrix is changed.
EXAMPLES::
sage: a = matrix(2,3,range(6)); a [0 1 2] [3 4 5] sage: a.row(0) (0, 1, 2) sage: a.row(1) (3, 4, 5) sage: a.row(-1) # last row (3, 4, 5)
TESTS::
sage: a = matrix(2,3,range(6)); a [0 1 2] [3 4 5] sage: a.row(2) Traceback (most recent call last): ... IndexError: row index out of range sage: a.row(-3) Traceback (most recent call last): ... IndexError: row index out of range """ raise IndexError("matrix has no rows") cdef Py_ssize_t j
########################################################################### # Building matrices out of other matrices, rows, or columns ########################################################################### def stack(self, bottom, subdivide=False): r""" Return a new matrix formed by appending the matrix (or vector) ``bottom`` below ``self``::
[ self ] [ bottom ]
INPUT:
- ``bottom`` - a matrix, vector or free module element, whose dimensions are compatible with ``self``.
- ``subdivide`` - default: ``False`` - request the resulting matrix to have a new subdivision, separating ``self`` from ``bottom``.
OUTPUT:
A new matrix formed by appending ``bottom`` beneath ``self``. If ``bottom`` is a vector (or free module element) then in this context it is appropriate to consider it as a row vector. (The code first converts a vector to a 1-row matrix.)
If ``subdivide`` is ``True`` then any row subdivisions for the two matrices are preserved, and a new subdivision is added between ``self`` and ``bottom``. If the column divisions are identical, then they are preserved, otherwise they are discarded. When ``subdivide`` is ``False`` there is no subdivision information in the result.
.. warning::
If ``subdivide`` is ``True`` then unequal column subdivisions will be discarded, since it would be ambiguous how to interpret them. If the subdivision behavior is not what you need, you can manage subdivisions yourself with methods like :meth:`~sage.matrix.matrix2.Matrix.subdivisions` and :meth:`~sage.matrix.matrix2.Matrix.subdivide`. You might also find :func:`~sage.matrix.constructor.block_matrix` or :func:`~sage.matrix.constructor.block_diagonal_matrix` useful and simpler in some instances.
EXAMPLES:
Stacking with a matrix. ::
sage: A = matrix(QQ, 4, 3, range(12)) sage: B = matrix(QQ, 3, 3, range(9)) sage: A.stack(B) [ 0 1 2] [ 3 4 5] [ 6 7 8] [ 9 10 11] [ 0 1 2] [ 3 4 5] [ 6 7 8]
Stacking with a vector. ::
sage: A = matrix(QQ, 3, 2, [0, 2, 4, 6, 8, 10]) sage: v = vector(QQ, 2, [100, 200]) sage: A.stack(v) [ 0 2] [ 4 6] [ 8 10] [100 200]
Errors are raised if the sizes are incompatible. ::
sage: A = matrix(RR, [[1, 2],[3, 4]]) sage: B = matrix(RR, [[10, 20, 30], [40, 50, 60]]) sage: A.stack(B) Traceback (most recent call last): ... TypeError: number of columns must be the same, not 2 and 3
sage: v = vector(RR, [100, 200, 300]) sage: A.stack(v) Traceback (most recent call last): ... TypeError: number of columns must be the same, not 2 and 3
Setting ``subdivide`` to ``True`` will, in its simplest form, add a subdivision between ``self`` and ``bottom``. ::
sage: A = matrix(QQ, 2, 5, range(10)) sage: B = matrix(QQ, 3, 5, range(15)) sage: A.stack(B, subdivide=True) [ 0 1 2 3 4] [ 5 6 7 8 9] [--------------] [ 0 1 2 3 4] [ 5 6 7 8 9] [10 11 12 13 14]
Row subdivisions are preserved by stacking, and enriched, if subdivisions are requested. (So multiple stackings can be recorded.) ::
sage: A = matrix(QQ, 2, 4, range(8)) sage: A.subdivide([1], None) sage: B = matrix(QQ, 3, 4, range(12)) sage: B.subdivide([2], None) sage: A.stack(B, subdivide=True) [ 0 1 2 3] [-----------] [ 4 5 6 7] [-----------] [ 0 1 2 3] [ 4 5 6 7] [-----------] [ 8 9 10 11]
Column subdivisions can be preserved, but only if they are identical. Otherwise, this information is discarded and must be managed separately. ::
sage: A = matrix(QQ, 2, 5, range(10)) sage: A.subdivide(None, [2,4]) sage: B = matrix(QQ, 3, 5, range(15)) sage: B.subdivide(None, [2,4]) sage: A.stack(B, subdivide=True) [ 0 1| 2 3| 4] [ 5 6| 7 8| 9] [-----+-----+--] [ 0 1| 2 3| 4] [ 5 6| 7 8| 9] [10 11|12 13|14]
sage: A.subdivide(None, [1,2]) sage: A.stack(B, subdivide=True) [ 0 1 2 3 4] [ 5 6 7 8 9] [--------------] [ 0 1 2 3 4] [ 5 6 7 8 9] [10 11 12 13 14]
The base ring of the result is the common parent for the base rings of ``self`` and ``bottom``. In particular, the parent for ``A.stack(B)`` and ``B.stack(A)`` should be equal::
sage: A = matrix(QQ, 1, 2, [1,2]) sage: B = matrix(RR, 1, 2, [sin(1.1), sin(2.2)]) sage: C = A.stack(B); C [ 1.00000000000000 2.00000000000000] [0.891207360061435 0.808496403819590] sage: C.parent() Full MatrixSpace of 2 by 2 dense matrices over Real Field with 53 bits of precision
sage: D = B.stack(A); D [0.891207360061435 0.808496403819590] [ 1.00000000000000 2.00000000000000] sage: D.parent() Full MatrixSpace of 2 by 2 dense matrices over Real Field with 53 bits of precision
::
sage: R.<y> = PolynomialRing(ZZ) sage: A = matrix(QQ, 1, 2, [1, 2/3]) sage: B = matrix(R, 1, 2, [y, y^2])
sage: C = A.stack(B); C [ 1 2/3] [ y y^2] sage: C.parent() Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in y over Rational Field
Stacking a dense matrix atop a sparse one returns a sparse matrix::
sage: M = Matrix(ZZ, 2, 3, range(6), sparse=False) sage: N = diagonal_matrix([10,11,12], sparse=True) sage: P = M.stack(N); P [ 0 1 2] [ 3 4 5] [10 0 0] [ 0 11 0] [ 0 0 12] sage: P.is_sparse() True sage: P = N.stack(M); P [10 0 0] [ 0 11 0] [ 0 0 12] [ 0 1 2] [ 3 4 5] sage: P.is_sparse() True
One can stack matrices over different rings (:trac:`16399`). ::
sage: M = Matrix(ZZ, 2, 3, range(6)) sage: N = Matrix(QQ, 1, 3, [10,11,12]) sage: M.stack(N) [ 0 1 2] [ 3 4 5] [10 11 12] sage: N.stack(M) [10 11 12] [ 0 1 2] [ 3 4 5]
TESTS:
A legacy test from the original implementation. ::
sage: M = Matrix(QQ, 2, 3, range(6)) sage: N = Matrix(QQ, 1, 3, [10,11,12]) sage: M.stack(N) [ 0 1 2] [ 3 4 5] [10 11 12]
Non-matrices fail gracefully::
sage: M.stack(polygen(QQ)) Traceback (most recent call last): ... TypeError: a matrix must be stacked with another matrix or a vector
AUTHORS:
- Rob Beezer (2011-03-19): rewritten to mirror code for :meth:`augment`
- Jeroen Demeyer (2015-01-06): refactor, see :trac:`16399`. Put all boilerplate in one place (here) and put the actual type-dependent implementation in ``_stack_impl``. """ cdef Matrix other else: else: 'another matrix or a vector')
# If one of the matrices is sparse, return a sparse matrix
cdef _stack_impl(self, bottom): """ Implementation of :meth:`stack`.
Assume that ``self`` and ``other`` are compatible in the sense that they have the same base ring and that both are either dense or sparse. """ cdef Matrix Z
cdef Py_ssize_t r, c
def augment(self, right, subdivide=False): r""" Returns a new matrix formed by appending the matrix (or vector) ``right`` on the right side of ``self``.
INPUT:
- ``right`` - a matrix, vector or free module element, whose dimensions are compatible with ``self``.
- ``subdivide`` - default: ``False`` - request the resulting matrix to have a new subdivision, separating ``self`` from ``right``.
OUTPUT:
A new matrix formed by appending ``right`` onto the right side of ``self``. If ``right`` is a vector (or free module element) then in this context it is appropriate to consider it as a column vector. (The code first converts a vector to a 1-column matrix.)
If ``subdivide`` is ``True`` then any column subdivisions for the two matrices are preserved, and a new subdivision is added between ``self`` and ``right``. If the row divisions are identical, then they are preserved, otherwise they are discarded. When ``subdivide`` is ``False`` there is no subdivision information in the result.
.. warning:: If ``subdivide`` is ``True`` then unequal row subdivisions will be discarded, since it would be ambiguous how to interpret them. If the subdivision behavior is not what you need, you can manage subdivisions yourself with methods like :meth:`~sage.matrix.matrix2.Matrix.get_subdivisions` and :meth:`~sage.matrix.matrix2.Matrix.subdivide`. You might also find :func:`~sage.matrix.constructor.block_matrix` or :func:`~sage.matrix.constructor.block_diagonal_matrix` useful and simpler in some instances.
EXAMPLES:
Augmenting with a matrix. ::
sage: A = matrix(QQ, 3, range(12)) sage: B = matrix(QQ, 3, range(9)) sage: A.augment(B) [ 0 1 2 3 0 1 2] [ 4 5 6 7 3 4 5] [ 8 9 10 11 6 7 8]
Augmenting with a vector. ::
sage: A = matrix(QQ, 2, [0, 2, 4, 6, 8, 10]) sage: v = vector(QQ, 2, [100, 200]) sage: A.augment(v) [ 0 2 4 100] [ 6 8 10 200]
Errors are raised if the sizes are incompatible. ::
sage: A = matrix(RR, [[1, 2],[3, 4]]) sage: B = matrix(RR, [[10, 20], [30, 40], [50, 60]]) sage: A.augment(B) Traceback (most recent call last): ... TypeError: number of rows must be the same, 2 != 3
sage: v = vector(RR, [100, 200, 300]) sage: A.augment(v) Traceback (most recent call last): ... TypeError: number of rows must be the same, 2 != 3
Setting ``subdivide`` to ``True`` will, in its simplest form, add a subdivision between ``self`` and ``right``. ::
sage: A = matrix(QQ, 3, range(12)) sage: B = matrix(QQ, 3, range(15)) sage: A.augment(B, subdivide=True) [ 0 1 2 3| 0 1 2 3 4] [ 4 5 6 7| 5 6 7 8 9] [ 8 9 10 11|10 11 12 13 14]
Column subdivisions are preserved by augmentation, and enriched, if subdivisions are requested. (So multiple augmentations can be recorded.) ::
sage: A = matrix(QQ, 3, range(6)) sage: A.subdivide(None, [1]) sage: B = matrix(QQ, 3, range(9)) sage: B.subdivide(None, [2]) sage: A.augment(B, subdivide=True) [0|1|0 1|2] [2|3|3 4|5] [4|5|6 7|8]
Row subdivisions can be preserved, but only if they are identical. Otherwise, this information is discarded and must be managed separately. ::
sage: A = matrix(QQ, 3, range(6)) sage: A.subdivide([1,3], None) sage: B = matrix(QQ, 3, range(9)) sage: B.subdivide([1,3], None) sage: A.augment(B, subdivide=True) [0 1|0 1 2] [---+-----] [2 3|3 4 5] [4 5|6 7 8] [---+-----]
sage: A.subdivide([1,2], None) sage: A.augment(B, subdivide=True) [0 1|0 1 2] [2 3|3 4 5] [4 5|6 7 8]
The result retains the base ring of ``self`` by coercing the elements of ``right`` into the base ring of ``self``. ::
sage: A = matrix(QQ, 2, [1,2]) sage: B = matrix(RR, 2, [sin(1.1), sin(2.2)]) sage: C = A.augment(B); C [ 1 183017397/205358938] [ 2 106580492/131825561] sage: C.parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: D = B.augment(A); D [0.89120736006... 1.00000000000000] [0.80849640381... 2.00000000000000] sage: D.parent() Full MatrixSpace of 2 by 2 dense matrices over Real Field with 53 bits of precision
Sometimes it is not possible to coerce into the base ring of ``self``. A solution is to change the base ring of ``self`` to a more expansive ring. Here we mix the rationals with a ring of polynomials with rational coefficients. ::
sage: R.<y> = PolynomialRing(QQ) sage: A = matrix(QQ, 1, [1,2]) sage: B = matrix(R, 1, [y, y^2])
sage: C = B.augment(A); C [ y y^2 1 2] sage: C.parent() Full MatrixSpace of 1 by 4 dense matrices over Univariate Polynomial Ring in y over Rational Field
sage: D = A.augment(B) Traceback (most recent call last): ... TypeError: not a constant polynomial
sage: E = A.change_ring(R) sage: F = E.augment(B); F [ 1 2 y y^2] sage: F.parent() Full MatrixSpace of 1 by 4 dense matrices over Univariate Polynomial Ring in y over Rational Field
AUTHORS:
- Naqi Jaffery (2006-01-24): examples - Rob Beezer (2010-12-07): vector argument, docstring, subdivisions """
else: raise TypeError("a matrix must be augmented with another matrix, " "or a vector")
cdef Matrix other
cdef Matrix Z
cdef Py_ssize_t r, c
def matrix_from_columns(self, columns): """ Return the matrix constructed from self using columns with indices in the columns list.
EXAMPLES::
sage: M = MatrixSpace(Integers(8),3,3) sage: A = M(range(9)); A [0 1 2] [3 4 5] [6 7 0] sage: A.matrix_from_columns([2,1]) [2 1] [5 4] [0 7] """
cdef Matrix A cdef Py_ssize_t ncols,k,r
raise IndexError("column %s out of range" % columns[i])
def delete_columns(self, dcols, check=True): """ Return the matrix constructed from deleting the columns with indices in the ``dcols`` list.
INPUT:
* ``dcols`` - list of indices of columns to be deleted from self. * ``check`` - checks whether any index in ``dcols`` is out of range. Defaults to ``True``.
.. SEEALSO::
The methods :meth:`delete_rows` and :meth:`matrix_from_columns` are related.
EXAMPLES::
sage: A = Matrix(3,4,range(12)); A [ 0 1 2 3] [ 4 5 6 7] [ 8 9 10 11] sage: A.delete_columns([0,2]) [ 1 3] [ 5 7] [ 9 11]
``dcols`` can be a tuple. But only the underlying set of indices matters. ::
sage: A.delete_columns((2,0,2)) [ 1 3] [ 5 7] [ 9 11]
The default is to check whether any index in ``dcols`` is out of range. ::
sage: A.delete_columns([-1,2,4]) Traceback (most recent call last): ... IndexError: [-1, 4] contains invalid indices sage: A.delete_columns([-1,2,4], check=False) [ 0 1 3] [ 4 5 7] [ 8 9 11]
TESTS:
The list of indices is checked. ::
sage: A.delete_columns("junk") Traceback (most recent call last): ... IndexError: ['j', 'k', 'n', 'u'] contains invalid indices
AUTHORS:
- Wai Yan Pong (2012-03-05) """
cdef list cols, diff_cols
def matrix_from_rows(self, rows): """ Return the matrix constructed from self using rows with indices in the rows list.
EXAMPLES::
sage: M = MatrixSpace(Integers(8),3,3) sage: A = M(range(9)); A [0 1 2] [3 4 5] [6 7 0] sage: A.matrix_from_rows([2,1]) [6 7 0] [3 4 5] """
cdef Matrix A cdef Py_ssize_t nrows,k,c
raise IndexError("row %s out of range" % rows[i])
def delete_rows(self, drows, check=True): """ Return the matrix constructed from deleting the rows with indices in the ``drows`` list.
INPUT:
* ``drows`` - list of indices of rows to be deleted from self. * ``check`` - checks whether any index in ``drows`` is out of range. Defaults to ``True``.
.. SEEALSO::
The methods :meth:`delete_columns` and :meth:`matrix_from_rows` are related.
EXAMPLES::
sage: A = Matrix(4,3,range(12)); A [ 0 1 2] [ 3 4 5] [ 6 7 8] [ 9 10 11] sage: A.delete_rows([0,2]) [ 3 4 5] [ 9 10 11]
``drows`` can be a tuple. But only the underlying set of indices matters. ::
sage: A.delete_rows((2,0,2)) [ 3 4 5] [ 9 10 11]
The default is to check whether the any index in ``drows`` is out of range. ::
sage: A.delete_rows([-1,2,4]) Traceback (most recent call last): ... IndexError: [-1, 4] contains invalid indices sage: A.delete_rows([-1,2,4], check=False) [ 0 1 2] [ 3 4 5] [ 9 10 11]
TESTS:
The list of indices is checked. ::
sage: A.delete_rows("junk") Traceback (most recent call last): ... IndexError: ['j', 'k', 'n', 'u'] contains invalid indices
AUTHORS
- Wai Yan Pong (2012-03-05) """
cdef list rows, diff_rows
def matrix_from_rows_and_columns(self, rows, columns): """ Return the matrix constructed from self from the given rows and columns.
EXAMPLES::
sage: M = MatrixSpace(Integers(8),3,3) sage: A = M(range(9)); A [0 1 2] [3 4 5] [6 7 0] sage: A.matrix_from_rows_and_columns([1], [0,2]) [3 5] sage: A.matrix_from_rows_and_columns([1,2], [1,2]) [4 5] [7 0]
Note that row and column indices can be reordered or repeated::
sage: A.matrix_from_rows_and_columns([2,1], [2,1]) [0 7] [5 4]
For example here we take from row 1 columns 2 then 0 twice, and do this 3 times.
::
sage: A.matrix_from_rows_and_columns([1,1,1],[2,0,0]) [5 3 3] [5 3 3] [5 3 3]
AUTHORS:
- Jaap Spies (2006-02-18)
- Didier Deshommes: some Pyrex speedups implemented """
rows = list(rows)
columns = list(columns)
cdef Matrix A cdef Py_ssize_t nrows, ncols,k,r,i,j
raise IndexError("column index out of range")
raise IndexError("row %s out of range" % i)
def submatrix(self, Py_ssize_t row=0, Py_ssize_t col=0, Py_ssize_t nrows=-1, Py_ssize_t ncols=-1): """ Return the matrix constructed from self using the specified range of rows and columns.
INPUT:
- ``row``, ``col`` - index of the starting row and column. Indices start at zero.
- ``nrows``, ``ncols`` - (optional) number of rows and columns to take. If not provided, take all rows below and all columns to the right of the starting entry.
.. SEEALSO::
The functions :func:`matrix_from_rows`, :func:`matrix_from_columns`, and :func:`matrix_from_rows_and_columns` allow one to select arbitrary subsets of rows and/or columns.
EXAMPLES:
Take the `3 \\times 3` submatrix starting from entry (1,1) in a `4 \\times 4` matrix::
sage: m = matrix(4, [1..16]) sage: m.submatrix(1, 1) [ 6 7 8] [10 11 12] [14 15 16]
Same thing, except take only two rows::
sage: m.submatrix(1, 1, 2) [ 6 7 8] [10 11 12]
And now take only one column::
sage: m.submatrix(1, 1, 2, 1) [ 6] [10]
You can take zero rows or columns if you want::
sage: m.submatrix(1, 1, 0) [] sage: parent(m.submatrix(1, 1, 0)) Full MatrixSpace of 0 by 3 dense matrices over Integer Ring """
def set_row(self, row, v): r""" Sets the entries of row ``row`` to the entries of ``v``.
INPUT:
- ``row`` - index of row to be set.
- ``v`` - a list or vector of the new entries.
OUTPUT:
Changes the matrix in-place, so there is no output.
EXAMPLES:
New entries may be contained in a vector.::
sage: A = matrix(QQ, 5, range(25)) sage: u = vector(QQ, [0, -1, -2, -3, -4]) sage: A.set_row(2, u) sage: A [ 0 1 2 3 4] [ 5 6 7 8 9] [ 0 -1 -2 -3 -4] [15 16 17 18 19] [20 21 22 23 24]
New entries may be in any sort of list.::
sage: A = matrix([[1, 2], [3, 4]]); A [1 2] [3 4] sage: A.set_row(0, [0, 0]); A [0 0] [3 4] sage: A.set_row(1, (0, 0)); A [0 0] [0 0]
TESTS::
sage: A = matrix([[1, 2], [3, 4]]) sage: A.set_row(2, [0, 0]); A Traceback (most recent call last): ... ValueError: row number must be between 0 and 1 (inclusive), not 2
sage: A.set_row(0, [0, 0, 0]) Traceback (most recent call last): ... ValueError: list of new entries must be of length 2 (not 3)
sage: A = matrix(2, [1, 2, 3, 4]) sage: A.set_row(0, [1/3, 1]); A Traceback (most recent call last): ... TypeError: Cannot set row with Rational Field elements over Integer Ring, use change_ring first. """
def set_column(self, col, v): r""" Sets the entries of column ``col`` to the entries of ``v``.
INPUT:
- ``col`` - index of column to be set.
- ``v`` - a list or vector of the new entries.
OUTPUT:
Changes the matrix in-place, so there is no output.
EXAMPLES:
New entries may be contained in a vector.::
sage: A = matrix(QQ, 5, range(25)) sage: u = vector(QQ, [0, -1, -2, -3, -4]) sage: A.set_column(2, u) sage: A [ 0 1 0 3 4] [ 5 6 -1 8 9] [10 11 -2 13 14] [15 16 -3 18 19] [20 21 -4 23 24]
New entries may be in any sort of list.::
sage: A = matrix([[1, 2], [3, 4]]); A [1 2] [3 4] sage: A.set_column(0, [0, 0]); A [0 2] [0 4] sage: A.set_column(1, (0, 0)); A [0 0] [0 0]
TESTS::
sage: A = matrix([[1, 2], [3, 4]]) sage: A.set_column(2, [0, 0]); A Traceback (most recent call last): ... ValueError: column number must be between 0 and 1 (inclusive), not 2
sage: A.set_column(0, [0, 0, 0]) Traceback (most recent call last): ... ValueError: list of new entries must be of length 2 (not 3)
sage: A = matrix(2, [1, 2, 3, 4]) sage: A.set_column(0, [1/4, 1]); A Traceback (most recent call last): ... TypeError: Cannot set column with Rational Field elements over Integer Ring, use change_ring first. """
#################################################################################### # Change of representation between dense and sparse. ####################################################################################
def dense_matrix(self): """ If this matrix is sparse, return a dense matrix with the same entries. If this matrix is dense, return this matrix (not a copy).
.. NOTE::
The definition of "dense" and "sparse" in Sage have nothing to do with the number of nonzero entries. Sparse and dense are properties of the underlying representation of the matrix.
EXAMPLES::
sage: A = MatrixSpace(QQ,2, sparse=True)([1,2,0,1]) sage: A.is_sparse() True sage: B = A.dense_matrix() sage: B.is_sparse() False sage: A*B [1 4] [0 1] sage: A.parent() Full MatrixSpace of 2 by 2 sparse matrices over Rational Field sage: B.parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field
In Sage, the product of a sparse and a dense matrix is always dense::
sage: (A*B).parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field sage: (B*A).parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field
TESTS:
Make sure that subdivisions are preserved when switching between dense and sparse matrices::
sage: a = matrix(ZZ, 3, range(9)) sage: a.subdivide([1,2],2) sage: a.subdivisions() ([1, 2], [2]) sage: b = a.sparse_matrix().dense_matrix() sage: b.subdivisions() ([1, 2], [2])
Ensure we can compute the correct dense matrix even if the dict items are ETuples (see :trac:`17658`)::
sage: from sage.rings.polynomial.polydict import ETuple sage: matrix(GF(5^2,"z"),{ETuple((1, 1)): 2}).dense_matrix() [0 0] [0 2] """ cdef Matrix A
def sparse_matrix(self): """ If this matrix is dense, return a sparse matrix with the same entries. If this matrix is sparse, return this matrix (not a copy).
.. NOTE::
The definition of "dense" and "sparse" in Sage have nothing to do with the number of nonzero entries. Sparse and dense are properties of the underlying representation of the matrix.
EXAMPLES::
sage: A = MatrixSpace(QQ,2, sparse=False)([1,2,0,1]) sage: A.is_sparse() False sage: B = A.sparse_matrix() sage: B.is_sparse() True sage: A [1 2] [0 1] sage: B [1 2] [0 1] sage: A*B [1 4] [0 1] sage: A.parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field sage: B.parent() Full MatrixSpace of 2 by 2 sparse matrices over Rational Field sage: (A*B).parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field sage: (B*A).parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field """
def matrix_space(self, nrows=None, ncols=None, sparse=None): """ Return the ambient matrix space of self.
INPUT:
- ``nrows``, ``ncols`` - (optional) number of rows and columns in returned matrix space.
- ``sparse`` - whether the returned matrix space uses sparse or dense matrices.
EXAMPLES::
sage: m = matrix(3, [1..9]) sage: m.matrix_space() Full MatrixSpace of 3 by 3 dense matrices over Integer Ring sage: m.matrix_space(ncols=2) Full MatrixSpace of 3 by 2 dense matrices over Integer Ring sage: m.matrix_space(1) Full MatrixSpace of 1 by 3 dense matrices over Integer Ring sage: m.matrix_space(1, 2, True) Full MatrixSpace of 1 by 2 sparse matrices over Integer Ring
sage: M = MatrixSpace(QQ, 3, implementation='generic') sage: m = M.an_element() sage: m.matrix_space() Full MatrixSpace of 3 by 3 dense matrices over Rational Field (using Matrix_generic_dense) sage: m.matrix_space(nrows=2, ncols=12) Full MatrixSpace of 2 by 12 dense matrices over Rational Field (using Matrix_generic_dense) sage: m.matrix_space(nrows=2, sparse=True) Full MatrixSpace of 2 by 3 sparse matrices over Rational Field """
else: else:
def new_matrix(self, nrows=None, ncols=None, entries=None, coerce=True, copy=True, sparse=None): """ Create a matrix in the parent of this matrix with the given number of rows, columns, etc. The default parameters are the same as for self.
INPUT:
These three variables get sent to :func:`matrix_space`:
- ``nrows``, ``ncols`` - number of rows and columns in returned matrix. If not specified, defaults to ``None`` and will give a matrix of the same size as self. - ``sparse`` - whether returned matrix is sparse or not. Defaults to same value as self.
The remaining three variables (``coerce``, ``entries``, and ``copy``) are used by :func:`sage.matrix.matrix_space.MatrixSpace` to construct the new matrix.
.. warning::
This function called with no arguments returns the zero matrix of the same dimension and sparseness of self.
EXAMPLES::
sage: A = matrix(ZZ,2,2,[1,2,3,4]); A [1 2] [3 4] sage: A.new_matrix() [0 0] [0 0] sage: A.new_matrix(1,1) [0] sage: A.new_matrix(3,3).parent() Full MatrixSpace of 3 by 3 dense matrices over Integer Ring
::
sage: A = matrix(RR,2,3,[1.1,2.2,3.3,4.4,5.5,6.6]); A [1.10000000000000 2.20000000000000 3.30000000000000] [4.40000000000000 5.50000000000000 6.60000000000000] sage: A.new_matrix() [0.000000000000000 0.000000000000000 0.000000000000000] [0.000000000000000 0.000000000000000 0.000000000000000] sage: A.new_matrix().parent() Full MatrixSpace of 2 by 3 dense matrices over Real Field with 53 bits of precision
::
sage: M = MatrixSpace(ZZ, 2, 3, implementation='generic') sage: m = M.an_element() sage: m.new_matrix().parent() Full MatrixSpace of 2 by 3 dense matrices over Integer Ring (using Matrix_generic_dense) sage: m.new_matrix(3,3).parent() Full MatrixSpace of 3 by 3 dense matrices over Integer Ring (using Matrix_generic_dense) sage: m.new_matrix(3,3, sparse=True).parent() Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring """
def block_sum(self, Matrix other): """ Return the block matrix that has self and other on the diagonal::
[ self 0 ] [ 0 other ]
EXAMPLES::
sage: A = matrix(QQ[['t']], 2, range(1, 5)) sage: A.block_sum(100*A) [ 1 2 0 0] [ 3 4 0 0] [ 0 0 100 200] [ 0 0 300 400] """ raise TypeError("other must be a Matrix") |