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matrix1.pyx
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matrix1.pyx
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"""
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.sequence cimport PySequence_Fast
import sage.modules.free_module
from sage.structure.coerce 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
"""
w = self.list()
cdef Py_ssize_t nr, nc, i, j
nr = self._nrows
nc = self._ncols
v = []
for i from 0 <= i < nr:
tmp = []
for j from 0 <= j < nc:
tmp.append(w[i*nc + j]._pari_init_())
v.append( ','.join(tmp))
return 'Mat([%s])'%(';'.join(v))
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
"""
from sage.libs.pari.all import pari
return pari.matrix(self._nrows, self._ncols, self._list())
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
v = []
for i from 0 <= i < self._nrows:
tmp = []
for j from 0 <= j < self._ncols:
tmp.append(self.get_unsafe(i,j)._gap_init_())
v.append( '[%s]'%(','.join(tmp)) )
# It is needed to multiply with 'One(...)', because
# otherwise the result would not be a gap matrix
return '[%s]*One(%s)'%(','.join(v),sage.interfaces.gap.gap(self.base_ring()).name())
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 ] ]
"""
from sage.libs.gap.libgap import libgap
return libgap._construct_matrix(self)
def _fricas_init_(self):
"""
Return a FriCAS string representation of this matrix.
EXAMPLES::
sage: M = matrix(ZZ,2,range(4))
sage: fricas(M) # optional - fricas
+0 1+
| |
+2 3+
::
sage: M = matrix(QQ,3,[1,2,3,4/3,5/3,6/4,7,8,9])
sage: fricas(M).sage().parent() # optional - fricas
Full MatrixSpace of 3 by 3 dense matrices over Rational Field
::
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: fricas(M) # optional - fricas
+ 2 +
|- 9 x - 2 x + 2 x - 1 |
| |
| 2 2 |
+ x + 8 x - 3 x + 5+
"""
s = str(self.rows()).replace('(','[').replace(')',']')
R = self.base_ring()
try:
R._fricas_()
except TypeError:
return "matrix(%s)"%(s)
return "matrix(%s)$Matrix(%s)"%(s, R._fricas_init_())
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]]
"""
s = str(self.rows()).replace('(','[').replace(')',']')
return "(%s)"%(s)
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
v = []
for i from 0 <= i < self._nrows:
tmp = []
for j from 0 <= j < self._ncols:
tmp.append(self.get_unsafe(i,j)._maxima_init_())
v.append( '[%s]'%(','.join(tmp)) )
return 'matrix(%s)'%(','.join(v))
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]}}'
"""
return '{' + ', '.join([v._mathematica_init_() for v in self.rows()]) + '}'
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
One sparse matrix::
sage: M = matrix(QQ,2,2,[4,6,55,0],sparse=True)
sage: T = magma(M); T # optional - magma
Sparse matrix with 2 rows and 2 columns over Rational Field
sage: T.Determinant() # optional - magma
-330
"""
if self.is_sparse():
R = magma(self.base_ring())
s = "SparseMatrix({}, {}, {}, ["
s = s.format(R.name(), self.nrows(), self.ncols())
entries = ("<{}, {}, {}>".format(ij[0] + 1, ij[1] + 1,
mij._magma_init_(magma))
for ij, mij in self.dict().items())
return s + ', '.join(entries) + "])"
else:
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.
"""
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()))
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 |
TESTS:
Entries of the matrix get promoted to the base ring (:trac:`28566`)::
sage: R.<x,y> = QQ[]
sage: m = macaulay2(matrix(R, [[1, 2], [3, 4]])) # optional - macaulay2
sage: m.ring()._operator('===', R).sage() # optional - macaulay2
True
"""
if macaulay2 is None:
from sage.interfaces.macaulay2 import macaulay2 as m2_default
macaulay2 = m2_default
entries = [list(row) for row in self]
return macaulay2(entries).matrix().promote(self.base_ring())
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)
"""
w = self.list()
cdef Py_ssize_t nr, nc, i, j
nr = self._nrows
nc = self._ncols
v = []
for i from 0 <= i < nr:
tmp = []
for j from 0 <= j < nc:
tmp.append(w[i*nc + j]._pari_init_())
v.append( ','.join(tmp))
return '[%s]'%(';'.join(v))
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)
"""
if self.is_sparse():
entries = list(self.dict().items())
entries.sort()
# We hand-format the keys to get rid of the space that would
# normally follow the comma
entries = [(sib.name('(%d,%d)'%k), sib(v, 2)) for k,v in entries]
return sib.name('matrix')(self.base_ring(),
sib.int(self.nrows()),
sib.int(self.ncols()),
sib.dict(entries))
elif self.nrows() == 0 or self.ncols() == 0:
return sib.name('matrix')(self.base_ring(),
sib.int(self.nrows()),
sib.int(self.ncols()))
else:
entries = [[sib(v, 2) for v in row] for row in self.rows()]
return sib.name('matrix')(self.base_ring(), entries)
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)
"""
import numpy
A = numpy.matrix(self.list(), dtype=dtype)
return numpy.resize(A,(self.nrows(), self.ncols()))
# 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
"""
return self.change_ring(self.base_ring().fraction_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
"""
if hasattr(self._base_ring, 'cover_ring'):
S = self._base_ring.cover_ring()
if S is not self._base_ring:
return self.change_ring(S)
return self
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
"""
try:
S = self._base_ring.cover_ring()
if S is not self._base_ring:
return self.parent().change_ring(S)([v.lift_centered() for v in self])
except AttributeError:
pass
return self
#############################################################################################
# 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
"""
if copy not in [True, False]:
msg = "'copy' must be True or False, not {0}"
raise ValueError(msg.format(copy))
x = self.fetch('columns')
if x is not None:
if copy: return list(x)
return x
if self.is_sparse():
columns = self.sparse_columns(copy=copy)
else:
columns = self.dense_columns(copy=copy)
self.cache('columns', columns)
if copy: return list(columns)
return columns
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
"""
if copy not in [True, False]:
msg = "'copy' must be True or False, not {0}"
raise ValueError(msg.format(copy))
x = self.fetch('rows')
if x is not None:
if copy: return list(x)
return x
if self.is_sparse():
rows = self.sparse_rows(copy=copy)
else:
rows = self.dense_rows(copy=copy)
self.cache('rows', rows)
if copy: return list(rows)
return rows
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]
"""
x = self.fetch('dense_columns')
if x is not None:
if copy: return list(x)
return x
cdef Py_ssize_t i
A = self if self.is_dense() else self.dense_matrix()
C = [A.column(i) for i in range(self._ncols)]
# Make the vectors immutable since we are caching them
for x in C:
x.set_immutable()
# cache result
self.cache('dense_columns', C)
if copy:
return list(C)
else:
return C
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]
"""
x = self.fetch('dense_rows')
if x is not None:
if copy: return list(x)
return x
cdef Py_ssize_t i
A = self if self.is_dense() else self.dense_matrix()
R = [A.row(i) for i in range(self._nrows)]
# Make the vectors immutable since we are caching them
for x in R:
x.set_immutable()
# cache result
self.cache('dense_rows', R)
if copy:
return list(R)
else:
return R
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]