Dense matrices using a NumPy backend

AUTHORS:

• Jason Grout, Sep 2008: switch to NumPy backend, factored out the Matrix_double_dense class

• Josh Kantor

• William Stein: many bug fixes and touch ups.

class sage.matrix.matrix_numpy_dense.Matrix_numpy_dense

Bases: Matrix_dense

Fill the matrix with entries.

The numpy matrix must have already been allocated.

INPUT:

• parent – a matrix space over RDF

• entries – see matrix()

• copy – ignored (for backwards compatibility)

• coerce – if True (the default), convert elements to the base ring before passing them to NumPy. If False, pass the elements to NumPy as given.

EXAMPLES:

Sage

sage: matrix(RDF,3,range(9))
[0.0 1.0 2.0]
[3.0 4.0 5.0]
[6.0 7.0 8.0]
sage: matrix(CDF,3,3,2)
[2.0 0.0 0.0]
[0.0 2.0 0.0]
[0.0 0.0 2.0]

Python

>>> from sage.all import *
>>> matrix(RDF,Integer(3),range(Integer(9)))
[0.0 1.0 2.0]
[3.0 4.0 5.0]
[6.0 7.0 8.0]
>>> matrix(CDF,Integer(3),Integer(3),Integer(2))
[2.0 0.0 0.0]
[0.0 2.0 0.0]
[0.0 0.0 2.0]

Sage Live

matrix(RDF,3,range(9))
matrix(CDF,3,3,2)

is_symmetric(tol=1e-12)

Return whether this matrix is symmetric, to the given tolerance.

EXAMPLES:

Sage

sage: m = matrix(RDF,2,2,range(4)); m
[0.0 1.0]
[2.0 3.0]
sage: m.is_symmetric()
False
sage: m[1,0] = 1.1; m
[0.0 1.0]
[1.1 3.0]
sage: m.is_symmetric()
False

Python

>>> from sage.all import *
>>> m = matrix(RDF,Integer(2),Integer(2),range(Integer(4))); m
[0.0 1.0]
[2.0 3.0]
>>> m.is_symmetric()
False
>>> m[Integer(1),Integer(0)] = RealNumber('1.1'); m
[0.0 1.0]
[1.1 3.0]
>>> m.is_symmetric()
False

Sage Live

m = matrix(RDF,2,2,range(4)); m
m.is_symmetric()
m[1,0] = 1.1; m
m.is_symmetric()

The tolerance inequality is strict:

Sage

sage: m.is_symmetric(tol=0.1)
False
sage: m.is_symmetric(tol=0.11)
True

Python

>>> from sage.all import *
>>> m.is_symmetric(tol=RealNumber('0.1'))
False
>>> m.is_symmetric(tol=RealNumber('0.11'))
True

Sage Live

m.is_symmetric(tol=0.1)
m.is_symmetric(tol=0.11)

numpy(dtype=None)

Return a copy of the matrix as a numpy array.

It uses the numpy C/api so is very fast.

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

sage: m = matrix(RDF,[[1,2],[3,4]])
sage: n = m.numpy()
sage: import numpy
sage: tuple(numpy.linalg.eig(n))
(array([-0.37228132...,  5.37228132...]),
 array([[-0.82456484..., -0.41597356...],
       [ 0.56576746..., -0.90937671...]]))
sage: m = matrix(RDF, 2, range(6)); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
sage: m.numpy()
array([[0., 1., 2.],
       [3., 4., 5.]])

Python

>>> from sage.all import *
>>> m = matrix(RDF,[[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
>>> n = m.numpy()
>>> import numpy
>>> tuple(numpy.linalg.eig(n))
(array([-0.37228132...,  5.37228132...]),
 array([[-0.82456484..., -0.41597356...],
       [ 0.56576746..., -0.90937671...]]))
>>> m = matrix(RDF, Integer(2), range(Integer(6))); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
>>> m.numpy()
array([[0., 1., 2.],
       [3., 4., 5.]])

Sage Live

m = matrix(RDF,[[1,2],[3,4]])
n = m.numpy()
import numpy
tuple(numpy.linalg.eig(n))
m = matrix(RDF, 2, range(6)); m
m.numpy()

Alternatively, numpy automatically calls this function (via the magic __array__ method) to convert Sage matrices to numpy arrays:

Sage

sage: import numpy
sage: m = matrix(RDF, 2, range(6)); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
sage: numpy.array(m)
array([[0., 1., 2.],
       [3., 4., 5.]])
sage: numpy.array(m).dtype
dtype('float64')
sage: m = matrix(CDF, 2, range(6)); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
sage: numpy.array(m)
array([[0.+0.j, 1.+0.j, 2.+0.j],
       [3.+0.j, 4.+0.j, 5.+0.j]])
sage: numpy.array(m).dtype
dtype('complex128')

Python

>>> from sage.all import *
>>> import numpy
>>> m = matrix(RDF, Integer(2), range(Integer(6))); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
>>> numpy.array(m)
array([[0., 1., 2.],
       [3., 4., 5.]])
>>> numpy.array(m).dtype
dtype('float64')
>>> m = matrix(CDF, Integer(2), range(Integer(6))); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
>>> numpy.array(m)
array([[0.+0.j, 1.+0.j, 2.+0.j],
       [3.+0.j, 4.+0.j, 5.+0.j]])
>>> numpy.array(m).dtype
dtype('complex128')

Sage Live

import numpy
m = matrix(RDF, 2, range(6)); m
numpy.array(m)
numpy.array(m).dtype
m = matrix(CDF, 2, range(6)); m
numpy.array(m)
numpy.array(m).dtype

transpose()

Return the transpose of this matrix, without changing self.

EXAMPLES:

Sage

sage: m = matrix(RDF,2,3,range(6)); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
sage: m2 = m.transpose()
sage: m[0,0] = 2
sage: m2           #note that m2 hasn't changed
[0.0 3.0]
[1.0 4.0]
[2.0 5.0]

Python

>>> from sage.all import *
>>> m = matrix(RDF,Integer(2),Integer(3),range(Integer(6))); m
[0.0 1.0 2.0]
[3.0 4.0 5.0]
>>> m2 = m.transpose()
>>> m[Integer(0),Integer(0)] = Integer(2)
>>> m2           #note that m2 hasn't changed
[0.0 3.0]
[1.0 4.0]
[2.0 5.0]

Sage Live

m = matrix(RDF,2,3,range(6)); m
m2 = m.transpose()
m[0,0] = 2
m2           #note that m2 hasn't changed

.T is a convenient shortcut for the transpose:

Sage

sage: m.T
[2.0 3.0]
[1.0 4.0]
[2.0 5.0]

sage: m = matrix(RDF,0,3); m
[]
sage: m.transpose()
[]
sage: m.transpose().parent()
Full MatrixSpace of 3 by 0 dense matrices over Real Double Field

Python

>>> from sage.all import *
>>> m.T
[2.0 3.0]
[1.0 4.0]
[2.0 5.0]

>>> m = matrix(RDF,Integer(0),Integer(3)); m
[]
>>> m.transpose()
[]
>>> m.transpose().parent()
Full MatrixSpace of 3 by 0 dense matrices over Real Double Field

Sage Live

m.T
m = matrix(RDF,0,3); m
m.transpose()
m.transpose().parent()

Make live