Symbolic dense matrices¶

EXAMPLES:

Sage

sage: matrix(SR, 2, 2, range(4))
[0 1]
[2 3]
sage: matrix(SR, 2, 2, var('t'))
[t 0]
[0 t]

Python

>>> from sage.all import *
>>> matrix(SR, Integer(2), Integer(2), range(Integer(4)))
[0 1]
[2 3]
>>> matrix(SR, Integer(2), Integer(2), var('t'))
[t 0]
[0 t]

Sage Live

matrix(SR, 2, 2, range(4))
matrix(SR, 2, 2, var('t'))

Arithmetic:

Sage

sage: -matrix(SR, 2, range(4))
[ 0 -1]
[-2 -3]
sage: m = matrix(SR, 2, [1..4]); sqrt(2)*m
[  sqrt(2) 2*sqrt(2)]
[3*sqrt(2) 4*sqrt(2)]
sage: m = matrix(SR, 4, [1..4^2])
sage: m * m
[ 90 100 110 120]
[202 228 254 280]
[314 356 398 440]
[426 484 542 600]

sage: m = matrix(SR, 3, [1, 2, 3]); m
[1]
[2]
[3]
sage: m.transpose() * m
[14]

Python

>>> from sage.all import *
>>> -matrix(SR, Integer(2), range(Integer(4)))
[ 0 -1]
[-2 -3]
>>> m = matrix(SR, Integer(2), (ellipsis_range(Integer(1),Ellipsis,Integer(4)))); sqrt(Integer(2))*m
[  sqrt(2) 2*sqrt(2)]
[3*sqrt(2) 4*sqrt(2)]
>>> m = matrix(SR, Integer(4), (ellipsis_range(Integer(1),Ellipsis,Integer(4)**Integer(2))))
>>> m * m
[ 90 100 110 120]
[202 228 254 280]
[314 356 398 440]
[426 484 542 600]

>>> m = matrix(SR, Integer(3), [Integer(1), Integer(2), Integer(3)]); m
[1]
[2]
[3]
>>> m.transpose() * m
[14]

Sage Live

-matrix(SR, 2, range(4))
m = matrix(SR, 2, [1..4]); sqrt(2)*m
m = matrix(SR, 4, [1..4^2])
m * m
m = matrix(SR, 3, [1, 2, 3]); m
m.transpose() * m

Computing inverses:

Sage

sage: M = matrix(SR, 2, var('a,b,c,d'))
sage: ~M
[1/a - b*c/(a^2*(b*c/a - d))           b/(a*(b*c/a - d))]
[          c/(a*(b*c/a - d))              -1/(b*c/a - d)]
sage: (~M*M).simplify_rational()
[1 0]
[0 1]
sage: M = matrix(SR, 3, 3, range(9)) - var('t')
sage: (~M * M).simplify_rational()
[1 0 0]
[0 1 0]
[0 0 1]

sage: matrix(SR, 1, 1, 1).inverse()
[1]
sage: matrix(SR, 0, 0).inverse()
[]
sage: matrix(SR, 3, 0).inverse()
Traceback (most recent call last):
...
ArithmeticError: self must be a square matrix

Python

>>> from sage.all import *
>>> M = matrix(SR, Integer(2), var('a,b,c,d'))
>>> ~M
[1/a - b*c/(a^2*(b*c/a - d))           b/(a*(b*c/a - d))]
[          c/(a*(b*c/a - d))              -1/(b*c/a - d)]
>>> (~M*M).simplify_rational()
[1 0]
[0 1]
>>> M = matrix(SR, Integer(3), Integer(3), range(Integer(9))) - var('t')
>>> (~M * M).simplify_rational()
[1 0 0]
[0 1 0]
[0 0 1]

>>> matrix(SR, Integer(1), Integer(1), Integer(1)).inverse()
[1]
>>> matrix(SR, Integer(0), Integer(0)).inverse()
[]
>>> matrix(SR, Integer(3), Integer(0)).inverse()
Traceback (most recent call last):
...
ArithmeticError: self must be a square matrix

Sage Live

M = matrix(SR, 2, var('a,b,c,d'))
~M
(~M*M).simplify_rational()
M = matrix(SR, 3, 3, range(9)) - var('t')
(~M * M).simplify_rational()
matrix(SR, 1, 1, 1).inverse()
matrix(SR, 0, 0).inverse()
matrix(SR, 3, 0).inverse()

Transposition:

Sage

sage: m = matrix(SR, 2, [sqrt(2), -1, pi, e^2])
sage: m.transpose()
[sqrt(2)      pi]
[     -1     e^2]

Python

>>> from sage.all import *
>>> m = matrix(SR, Integer(2), [sqrt(Integer(2)), -Integer(1), pi, e**Integer(2)])
>>> m.transpose()
[sqrt(2)      pi]
[     -1     e^2]

Sage Live

m = matrix(SR, 2, [sqrt(2), -1, pi, e^2])
m.transpose()

.T is a convenient shortcut for the transpose:

Sage

sage: m.T
[sqrt(2)      pi]
[     -1     e^2]

Python

>>> from sage.all import *
>>> m.T
[sqrt(2)      pi]
[     -1     e^2]

Sage Live

m.T

Test pickling:

Sage

sage: m = matrix(SR, 2, [sqrt(2), 3, pi, e]); m
[sqrt(2)       3]
[     pi       e]
sage: TestSuite(m).run()

Python

>>> from sage.all import *
>>> m = matrix(SR, Integer(2), [sqrt(Integer(2)), Integer(3), pi, e]); m
[sqrt(2)       3]
[     pi       e]
>>> TestSuite(m).run()

Sage Live

m = matrix(SR, 2, [sqrt(2), 3, pi, e]); m
TestSuite(m).run()

Comparison:

Sage

sage: m = matrix(SR, 2, [sqrt(2), 3, pi, e])
sage: m == m
True
sage: m != 3
True
sage: m = matrix(SR,2,[1..4]); n = m^2
sage: (exp(m+n) - exp(m)*exp(n)).simplify_rational() == 0       # indirect test
True

Python

>>> from sage.all import *
>>> m = matrix(SR, Integer(2), [sqrt(Integer(2)), Integer(3), pi, e])
>>> m == m
True
>>> m != Integer(3)
True
>>> m = matrix(SR,Integer(2),(ellipsis_range(Integer(1),Ellipsis,Integer(4)))); n = m**Integer(2)
>>> (exp(m+n) - exp(m)*exp(n)).simplify_rational() == Integer(0)       # indirect test
True

Sage Live

m = matrix(SR, 2, [sqrt(2), 3, pi, e])
m == m
m != 3
m = matrix(SR,2,[1..4]); n = m^2
(exp(m+n) - exp(m)*exp(n)).simplify_rational() == 0       # indirect test

Determinant:

Sage

sage: M = matrix(SR, 2, 2, [x,2,3,4])
sage: M.determinant()
4*x - 6
sage: M = matrix(SR, 3,3,range(9))
sage: M.det()
0
sage: t = var('t')
sage: M = matrix(SR, 2, 2, [cos(t), sin(t), -sin(t), cos(t)])
sage: M.det()
cos(t)^2 + sin(t)^2
sage: M = matrix([[sqrt(x),0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1]])
sage: det(M)
sqrt(x)

Python

>>> from sage.all import *
>>> M = matrix(SR, Integer(2), Integer(2), [x,Integer(2),Integer(3),Integer(4)])
>>> M.determinant()
4*x - 6
>>> M = matrix(SR, Integer(3),Integer(3),range(Integer(9)))
>>> M.det()
0
>>> t = var('t')
>>> M = matrix(SR, Integer(2), Integer(2), [cos(t), sin(t), -sin(t), cos(t)])
>>> M.det()
cos(t)^2 + sin(t)^2
>>> M = matrix([[sqrt(x),Integer(0),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(0),Integer(1)]])
>>> det(M)
sqrt(x)

Sage Live

M = matrix(SR, 2, 2, [x,2,3,4])
M.determinant()
M = matrix(SR, 3,3,range(9))
M.det()
t = var('t')
M = matrix(SR, 2, 2, [cos(t), sin(t), -sin(t), cos(t)])
M.det()
M = matrix([[sqrt(x),0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,1]])
det(M)

Permanents:

Sage

sage: M = matrix(SR, 2, 2, [x,2,3,4])
sage: M.permanent()
4*x + 6

Python

>>> from sage.all import *
>>> M = matrix(SR, Integer(2), Integer(2), [x,Integer(2),Integer(3),Integer(4)])
>>> M.permanent()
4*x + 6

Sage Live

M = matrix(SR, 2, 2, [x,2,3,4])
M.permanent()

Rank:

Sage

sage: M = matrix(SR, 5, 5, range(25))
sage: M.rank()
2
sage: M = matrix(SR, 5, 5, range(25)) - var('t')
sage: M.rank()
5

.. warning::

    :meth:`rank` may return the wrong answer if it cannot determine that a
    matrix element that is equivalent to zero is indeed so.

Python

>>> from sage.all import *
>>> M = matrix(SR, Integer(5), Integer(5), range(Integer(25)))
>>> M.rank()
2
>>> M = matrix(SR, Integer(5), Integer(5), range(Integer(25))) - var('t')
>>> M.rank()
5

.. warning::

    :meth:`rank` may return the wrong answer if it cannot determine that a
    matrix element that is equivalent to zero is indeed so.

Sage Live

M = matrix(SR, 5, 5, range(25))
M.rank()
M = matrix(SR, 5, 5, range(25)) - var('t')
M.rank()

Copying symbolic matrices:

Sage

sage: m = matrix(SR, 2, [sqrt(2), 3, pi, e])
sage: n = copy(m)
sage: n[0,0] = sin(1)
sage: m
[sqrt(2)       3]
[     pi       e]
sage: n
[sin(1)      3]
[    pi      e]

Python

>>> from sage.all import *
>>> m = matrix(SR, Integer(2), [sqrt(Integer(2)), Integer(3), pi, e])
>>> n = copy(m)
>>> n[Integer(0),Integer(0)] = sin(Integer(1))
>>> m
[sqrt(2)       3]
[     pi       e]
>>> n
[sin(1)      3]
[    pi      e]

Sage Live

m = matrix(SR, 2, [sqrt(2), 3, pi, e])
n = copy(m)
n[0,0] = sin(1)
m
n

Conversion to Maxima:

Sage

sage: m = matrix(SR, 2, [sqrt(2), 3, pi, e])
sage: m._maxima_()
matrix([sqrt(2),3],[%pi,%e])

Python

>>> from sage.all import *
>>> m = matrix(SR, Integer(2), [sqrt(Integer(2)), Integer(3), pi, e])
>>> m._maxima_()
matrix([sqrt(2),3],[%pi,%e])

Sage Live

m = matrix(SR, 2, [sqrt(2), 3, pi, e])
m._maxima_()

class sage.matrix.matrix_symbolic_dense.Matrix_symbolic_dense[source]¶

Bases: Matrix_generic_dense

arguments()[source]¶

Return a tuple of the arguments that self can take.

EXAMPLES:

Sage

sage: var('x,y,z')
(x, y, z)
sage: M = MatrixSpace(SR,2,2)
sage: M(x).arguments()
(x,)
sage: M(x+sin(x)).arguments()
(x,)

Python

>>> from sage.all import *
>>> var('x,y,z')
(x, y, z)
>>> M = MatrixSpace(SR,Integer(2),Integer(2))
>>> M(x).arguments()
(x,)
>>> M(x+sin(x)).arguments()
(x,)

Sage Live

var('x,y,z')
M = MatrixSpace(SR,2,2)
M(x).arguments()
M(x+sin(x)).arguments()

canonicalize_radical()[source]¶

Choose a canonical branch of each entry of self by calling Expression.canonicalize_radical() componentwise.

EXAMPLES:

Sage

sage: var('x','y')
(x, y)
sage: l1 = [sqrt(2)*sqrt(3)*sqrt(6) , log(x*y)]
sage: l2 = [sin(x/(x^2 + x)) , 1]
sage: m = matrix([l1, l2])
sage: m
[sqrt(6)*sqrt(3)*sqrt(2)                log(x*y)]
[       sin(x/(x^2 + x))                       1]
sage: m.canonicalize_radical()
[              6 log(x) + log(y)]
[ sin(1/(x + 1))               1]

Python

>>> from sage.all import *
>>> var('x','y')
(x, y)
>>> l1 = [sqrt(Integer(2))*sqrt(Integer(3))*sqrt(Integer(6)) , log(x*y)]
>>> l2 = [sin(x/(x**Integer(2) + x)) , Integer(1)]
>>> m = matrix([l1, l2])
>>> m
[sqrt(6)*sqrt(3)*sqrt(2)                log(x*y)]
[       sin(x/(x^2 + x))                       1]
>>> m.canonicalize_radical()
[              6 log(x) + log(y)]
[ sin(1/(x + 1))               1]

Sage Live

var('x','y')
l1 = [sqrt(2)*sqrt(3)*sqrt(6) , log(x*y)]
l2 = [sin(x/(x^2 + x)) , 1]
m = matrix([l1, l2])
m
m.canonicalize_radical()

charpoly(var='x', algorithm=None)[source]¶

Compute the characteristic polynomial of self, using maxima.

Note

The characteristic polynomial is defined as \(\det(xI-A)\).

INPUT:

var – (default: 'x') name of variable of charpoly

EXAMPLES:

Sage

sage: M = matrix(SR, 2, 2, var('a,b,c,d'))
sage: M.charpoly('t')
t^2 + (-a - d)*t - b*c + a*d
sage: matrix(SR, 5, [1..5^2]).charpoly()
x^5 - 65*x^4 - 250*x^3

Python

>>> from sage.all import *
>>> M = matrix(SR, Integer(2), Integer(2), var('a,b,c,d'))
>>> M.charpoly('t')
t^2 + (-a - d)*t - b*c + a*d
>>> matrix(SR, Integer(5), (ellipsis_range(Integer(1),Ellipsis,Integer(5)**Integer(2)))).charpoly()
x^5 - 65*x^4 - 250*x^3

Sage Live

M = matrix(SR, 2, 2, var('a,b,c,d'))
M.charpoly('t')
matrix(SR, 5, [1..5^2]).charpoly()

echelonize(**kwds)[source]¶: Echelonize using the classical algorithm.

eigenvalues(extend=True)[source]¶

Compute the eigenvalues by solving the characteristic polynomial in maxima.

The argument extend is ignored but kept for compatibility with other matrix classes.

EXAMPLES:

Sage

sage: a=matrix(SR,[[1,2],[3,4]])
sage: a.eigenvalues()
[-1/2*sqrt(33) + 5/2, 1/2*sqrt(33) + 5/2]

Python

>>> from sage.all import *
>>> a=matrix(SR,[[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
>>> a.eigenvalues()
[-1/2*sqrt(33) + 5/2, 1/2*sqrt(33) + 5/2]

Sage Live

a=matrix(SR,[[1,2],[3,4]])
a.eigenvalues()

eigenvectors_left(other=None)[source]¶

Compute the left eigenvectors of a matrix.

INPUT:

other – a square matrix \(B\) (default: None) in a generalized eigenvalue problem; if None, an ordinary eigenvalue problem is solved (currently supported only if the base ring of self is RDF or CDF)

OUTPUT:

For each distinct eigenvalue, returns a list of the form (e,V,n) where e is the eigenvalue, V is a list of eigenvectors forming a basis for the corresponding left eigenspace, and n is the algebraic multiplicity of the eigenvalue.

EXAMPLES:

Sage

sage: A = matrix(SR,3,3,range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: es = A.eigenvectors_left(); es
[(-3*sqrt(6) + 6, [(1, -1/5*sqrt(6) + 4/5, -2/5*sqrt(6) + 3/5)], 1),
 (3*sqrt(6) + 6, [(1, 1/5*sqrt(6) + 4/5, 2/5*sqrt(6) + 3/5)], 1),
 (0, [(1, -2, 1)], 1)]
sage: eval, [evec], mult = es[0]
sage: delta = eval*evec - evec*A
sage: abs(abs(delta)) < 1e-10
3/5*sqrt(((2*sqrt(6) - 3)*(sqrt(6) - 2) + 7*sqrt(6) - 18)^2 + ((sqrt(6) - 2)*(sqrt(6) - 4) + 6*sqrt(6) - 14)^2) < (1.00000000000000e-10)
sage: abs(abs(delta)).n() < 1e-10
True

Python

>>> from sage.all import *
>>> A = matrix(SR,Integer(3),Integer(3),range(Integer(9))); A
[0 1 2]
[3 4 5]
[6 7 8]
>>> es = A.eigenvectors_left(); es
[(-3*sqrt(6) + 6, [(1, -1/5*sqrt(6) + 4/5, -2/5*sqrt(6) + 3/5)], 1),
 (3*sqrt(6) + 6, [(1, 1/5*sqrt(6) + 4/5, 2/5*sqrt(6) + 3/5)], 1),
 (0, [(1, -2, 1)], 1)]
>>> eval, [evec], mult = es[Integer(0)]
>>> delta = eval*evec - evec*A
>>> abs(abs(delta)) < RealNumber('1e-10')
3/5*sqrt(((2*sqrt(6) - 3)*(sqrt(6) - 2) + 7*sqrt(6) - 18)^2 + ((sqrt(6) - 2)*(sqrt(6) - 4) + 6*sqrt(6) - 14)^2) < (1.00000000000000e-10)
>>> abs(abs(delta)).n() < RealNumber('1e-10')
True

Sage Live

A = matrix(SR,3,3,range(9)); A
es = A.eigenvectors_left(); es
eval, [evec], mult = es[0]
delta = eval*evec - evec*A
abs(abs(delta)) < 1e-10
abs(abs(delta)).n() < 1e-10

Sage

sage: A = matrix(SR, 2, 2, var('a,b,c,d'))
sage: A.eigenvectors_left()
[(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d - sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1)]
sage: es = A.eigenvectors_left(); es
[(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d - sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1)]
sage: eval, [evec], mult = es[0]
sage: delta = eval*evec - evec*A
sage: delta.apply_map(lambda x: x.full_simplify())
(0, 0)

Python

>>> from sage.all import *
>>> A = matrix(SR, Integer(2), Integer(2), var('a,b,c,d'))
>>> A.eigenvectors_left()
[(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d - sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1)]
>>> es = A.eigenvectors_left(); es
[(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1), (1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d - sqrt(a^2 + 4*b*c - 2*a*d + d^2))/c)], 1)]
>>> eval, [evec], mult = es[Integer(0)]
>>> delta = eval*evec - evec*A
>>> delta.apply_map(lambda x: x.full_simplify())
(0, 0)

Sage Live

A = matrix(SR, 2, 2, var('a,b,c,d'))
A.eigenvectors_left()
es = A.eigenvectors_left(); es
eval, [evec], mult = es[0]
delta = eval*evec - evec*A
delta.apply_map(lambda x: x.full_simplify())

This routine calls Maxima and can struggle with even small matrices with a few variables, such as a \(3\times 3\) matrix with three variables. However, if the entries are integers or rationals it can produce exact values in a reasonable time. These examples create 0-1 matrices from the adjacency matrices of graphs and illustrate how the format and type of the results differ when the base ring changes. First for matrices over the rational numbers, then the same matrix but viewed as a symbolic matrix.

Sage

sage: G=graphs.CycleGraph(5)
sage: am = G.adjacency_matrix()
sage: spectrum = am.eigenvectors_left()
sage: qqbar_evalue = spectrum[2][0]
sage: type(qqbar_evalue)
<class 'sage.rings.qqbar.AlgebraicNumber'>
sage: qqbar_evalue
0.618033988749895?

sage: am = G.adjacency_matrix().change_ring(SR)
sage: spectrum = am.eigenvectors_left()
sage: symbolic_evalue = spectrum[2][0]
sage: type(symbolic_evalue)
<class 'sage.symbolic.expression.Expression'>
sage: symbolic_evalue
1/2*sqrt(5) - 1/2

sage: bool(qqbar_evalue == symbolic_evalue)
True

Python

>>> from sage.all import *
>>> G=graphs.CycleGraph(Integer(5))
>>> am = G.adjacency_matrix()
>>> spectrum = am.eigenvectors_left()
>>> qqbar_evalue = spectrum[Integer(2)][Integer(0)]
>>> type(qqbar_evalue)
<class 'sage.rings.qqbar.AlgebraicNumber'>
>>> qqbar_evalue
0.618033988749895?

>>> am = G.adjacency_matrix().change_ring(SR)
>>> spectrum = am.eigenvectors_left()
>>> symbolic_evalue = spectrum[Integer(2)][Integer(0)]
>>> type(symbolic_evalue)
<class 'sage.symbolic.expression.Expression'>
>>> symbolic_evalue
1/2*sqrt(5) - 1/2

>>> bool(qqbar_evalue == symbolic_evalue)
True

Sage Live

G=graphs.CycleGraph(5)
am = G.adjacency_matrix()
spectrum = am.eigenvectors_left()
qqbar_evalue = spectrum[2][0]
type(qqbar_evalue)
qqbar_evalue
am = G.adjacency_matrix().change_ring(SR)
spectrum = am.eigenvectors_left()
symbolic_evalue = spectrum[2][0]
type(symbolic_evalue)
symbolic_evalue
bool(qqbar_evalue == symbolic_evalue)

A slightly larger matrix with a “nice” spectrum.

Sage

sage: G = graphs.CycleGraph(6)
sage: am = G.adjacency_matrix().change_ring(SR)
sage: am.eigenvectors_left()
[(-1, [(1, 0, -1, 1, 0, -1), (0, 1, -1, 0, 1, -1)], 2), (1, [(1, 0, -1, -1, 0, 1), (0, 1, 1, 0, -1, -1)], 2), (-2, [(1, -1, 1, -1, 1, -1)], 1), (2, [(1, 1, 1, 1, 1, 1)], 1)]

Python

>>> from sage.all import *
>>> G = graphs.CycleGraph(Integer(6))
>>> am = G.adjacency_matrix().change_ring(SR)
>>> am.eigenvectors_left()
[(-1, [(1, 0, -1, 1, 0, -1), (0, 1, -1, 0, 1, -1)], 2), (1, [(1, 0, -1, -1, 0, 1), (0, 1, 1, 0, -1, -1)], 2), (-2, [(1, -1, 1, -1, 1, -1)], 1), (2, [(1, 1, 1, 1, 1, 1)], 1)]

Sage Live

G = graphs.CycleGraph(6)
am = G.adjacency_matrix().change_ring(SR)
am.eigenvectors_left()

eigenvectors_right(other=None)[source]¶

Compute the right eigenvectors of a matrix.

INPUT:

other – a square matrix \(B\) (default: None) in a generalized eigenvalue problem; if None, an ordinary eigenvalue problem is solved (currently supported only if the base ring of self is RDF or CDF)

OUTPUT:

For each distinct eigenvalue, returns a list of the form (e,V,n) where e is the eigenvalue, V is a list of eigenvectors forming a basis for the corresponding right eigenspace, and n is the algebraic multiplicity of the eigenvalue.

EXAMPLES:

Sage

sage: A = matrix(SR,2,2,range(4)); A
[0 1]
[2 3]
sage: right = A.eigenvectors_right(); right
[(-1/2*sqrt(17) + 3/2, [(1, -1/2*sqrt(17) + 3/2)], 1), (1/2*sqrt(17) + 3/2, [(1, 1/2*sqrt(17) + 3/2)], 1)]

Python

>>> from sage.all import *
>>> A = matrix(SR,Integer(2),Integer(2),range(Integer(4))); A
[0 1]
[2 3]
>>> right = A.eigenvectors_right(); right
[(-1/2*sqrt(17) + 3/2, [(1, -1/2*sqrt(17) + 3/2)], 1), (1/2*sqrt(17) + 3/2, [(1, 1/2*sqrt(17) + 3/2)], 1)]

Sage Live

A = matrix(SR,2,2,range(4)); A
right = A.eigenvectors_right(); right

The right eigenvectors are nothing but the left eigenvectors of the transpose matrix:

Sage

sage: left  = A.transpose().eigenvectors_left(); left
[(-1/2*sqrt(17) + 3/2, [(1, -1/2*sqrt(17) + 3/2)], 1), (1/2*sqrt(17) + 3/2, [(1, 1/2*sqrt(17) + 3/2)], 1)]
sage: right[0][1] == left[0][1]
True

Python

>>> from sage.all import *
>>> left  = A.transpose().eigenvectors_left(); left
[(-1/2*sqrt(17) + 3/2, [(1, -1/2*sqrt(17) + 3/2)], 1), (1/2*sqrt(17) + 3/2, [(1, 1/2*sqrt(17) + 3/2)], 1)]
>>> right[Integer(0)][Integer(1)] == left[Integer(0)][Integer(1)]
True

Sage Live

left  = A.transpose().eigenvectors_left(); left
right[0][1] == left[0][1]

exp()[source]¶

Return the matrix exponential of this matrix \(X\), which is the matrix

\[e^X = \sum_{k=0}^{\infty} \frac{X^k}{k!}.\]

This function depends on maxima’s matrix exponentiation function, which does not deal well with floating point numbers. If the matrix has floating point numbers, they will be rounded automatically to rational numbers during the computation.

EXAMPLES:

Sage

sage: m = matrix(SR,2, [0,x,x,0]); m
[0 x]
[x 0]
sage: m.exp()
[1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)]
[1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)]
sage: exp(m)
[1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)]
[1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)]

Python

>>> from sage.all import *
>>> m = matrix(SR,Integer(2), [Integer(0),x,x,Integer(0)]); m
[0 x]
[x 0]
>>> m.exp()
[1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)]
[1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)]
>>> exp(m)
[1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)]
[1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)]

Sage Live

m = matrix(SR,2, [0,x,x,0]); m
m.exp()
exp(m)

Exponentiation works on 0x0 and 1x1 matrices, but the 1x1 example requires a patched version of maxima (Issue #32898) for now:

Sage

sage: m = matrix(SR,0,[]); m
[]
sage: m.exp()
[]
sage: m = matrix(SR,1,[2]); m
[2]
sage: m.exp()  # not tested, requires patched maxima
[e^2]

Python

>>> from sage.all import *
>>> m = matrix(SR,Integer(0),[]); m
[]
>>> m.exp()
[]
>>> m = matrix(SR,Integer(1),[Integer(2)]); m
[2]
>>> m.exp()  # not tested, requires patched maxima
[e^2]

Sage Live

m = matrix(SR,0,[]); m
m.exp()
m = matrix(SR,1,[2]); m
m.exp()  # not tested, requires patched maxima

Commuting matrices \(m, n\) have the property that \(e^{m+n} = e^m e^n\) (but non-commuting matrices need not):

Sage

sage: m = matrix(SR,2,[1..4]); n = m^2
sage: m*n
[ 37  54]
[ 81 118]
sage: n*m
[ 37  54]
[ 81 118]

sage: a = exp(m+n) - exp(m)*exp(n)
sage: a.simplify_rational() == 0
True

Python

>>> from sage.all import *
>>> m = matrix(SR,Integer(2),(ellipsis_range(Integer(1),Ellipsis,Integer(4)))); n = m**Integer(2)
>>> m*n
[ 37  54]
[ 81 118]
>>> n*m
[ 37  54]
[ 81 118]

>>> a = exp(m+n) - exp(m)*exp(n)
>>> a.simplify_rational() == Integer(0)
True

Sage Live

m = matrix(SR,2,[1..4]); n = m^2
m*n
n*m
a = exp(m+n) - exp(m)*exp(n)
a.simplify_rational() == 0

The input matrix must be square:

Sage

sage: m = matrix(SR,2,3,[1..6]); exp(m)
Traceback (most recent call last):
...
ValueError: exp only defined on square matrices

Python

>>> from sage.all import *
>>> m = matrix(SR,Integer(2),Integer(3),(ellipsis_range(Integer(1),Ellipsis,Integer(6)))); exp(m)
Traceback (most recent call last):
...
ValueError: exp only defined on square matrices

Sage Live

m = matrix(SR,2,3,[1..6]); exp(m)

In this example we take the symbolic answer and make it numerical at the end:

Sage

sage: exp(matrix(SR, [[1.2, 5.6], [3,4]])).change_ring(RDF)  # rel tol 1e-15
[ 346.5574872980695  661.7345909344504]
[354.50067371488416  677.4247827652946]

Python

>>> from sage.all import *
>>> exp(matrix(SR, [[RealNumber('1.2'), RealNumber('5.6')], [Integer(3),Integer(4)]])).change_ring(RDF)  # rel tol 1e-15
[ 346.5574872980695  661.7345909344504]
[354.50067371488416  677.4247827652946]

Sage Live

exp(matrix(SR, [[1.2, 5.6], [3,4]])).change_ring(RDF)  # rel tol 1e-15

Another example involving the reversed identity matrix, which we clumsily create:

Sage

sage: m = identity_matrix(SR,4); m = matrix(list(reversed(m.rows()))) * x
sage: exp(m)
[1/2*(e^(2*x) + 1)*e^(-x)                        0                        0 1/2*(e^(2*x) - 1)*e^(-x)]
[                       0 1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)                        0]
[                       0 1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)                        0]
[1/2*(e^(2*x) - 1)*e^(-x)                        0                        0 1/2*(e^(2*x) + 1)*e^(-x)]

Python

>>> from sage.all import *
>>> m = identity_matrix(SR,Integer(4)); m = matrix(list(reversed(m.rows()))) * x
>>> exp(m)
[1/2*(e^(2*x) + 1)*e^(-x)                        0                        0 1/2*(e^(2*x) - 1)*e^(-x)]
[                       0 1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)                        0]
[                       0 1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)                        0]
[1/2*(e^(2*x) - 1)*e^(-x)                        0                        0 1/2*(e^(2*x) + 1)*e^(-x)]

Sage Live

m = identity_matrix(SR,4); m = matrix(list(reversed(m.rows()))) * x
exp(m)

expand()[source]¶

Operate point-wise on each element.

EXAMPLES:

Sage

sage: M = matrix(2, 2, range(4)) - var('x')
sage: M*M
[      x^2 + 2      -2*x + 3]
[     -4*x + 6 (x - 3)^2 + 2]
sage: (M*M).expand()
[       x^2 + 2       -2*x + 3]
[      -4*x + 6 x^2 - 6*x + 11]

Python

>>> from sage.all import *
>>> M = matrix(Integer(2), Integer(2), range(Integer(4))) - var('x')
>>> M*M
[      x^2 + 2      -2*x + 3]
[     -4*x + 6 (x - 3)^2 + 2]
>>> (M*M).expand()
[       x^2 + 2       -2*x + 3]
[      -4*x + 6 x^2 - 6*x + 11]

Sage Live

M = matrix(2, 2, range(4)) - var('x')
M*M
(M*M).expand()

factor()[source]¶

Operate point-wise on each element.

EXAMPLES:

Sage

sage: M = matrix(SR, 2, 2, x^2 - 2*x + 1); M
[x^2 - 2*x + 1             0]
[            0 x^2 - 2*x + 1]
sage: M.factor()
[(x - 1)^2         0]
[        0 (x - 1)^2]

Python

>>> from sage.all import *
>>> M = matrix(SR, Integer(2), Integer(2), x**Integer(2) - Integer(2)*x + Integer(1)); M
[x^2 - 2*x + 1             0]
[            0 x^2 - 2*x + 1]
>>> M.factor()
[(x - 1)^2         0]
[        0 (x - 1)^2]

Sage Live

M = matrix(SR, 2, 2, x^2 - 2*x + 1); M
M.factor()

fcp(var='x')[source]¶

Return the factorization of the characteristic polynomial of self.

INPUT:

var – (default: 'x') name of variable of charpoly

EXAMPLES:

Sage

sage: a = matrix(SR,[[1,2],[3,4]])
sage: a.fcp()
x^2 - 5*x - 2
sage: [i for i in a.fcp()]
[(x^2 - 5*x - 2, 1)]
sage: a = matrix(SR,[[1,0],[0,2]])
sage: a.fcp()
(x - 2) * (x - 1)
sage: [i for i in a.fcp()]
[(x - 2, 1), (x - 1, 1)]
sage: a = matrix(SR, 5, [1..5^2])
sage: a.fcp()
(x^2 - 65*x - 250) * x^3
sage: list(a.fcp())
[(x^2 - 65*x - 250, 1), (x, 3)]

Python

>>> from sage.all import *
>>> a = matrix(SR,[[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
>>> a.fcp()
x^2 - 5*x - 2
>>> [i for i in a.fcp()]
[(x^2 - 5*x - 2, 1)]
>>> a = matrix(SR,[[Integer(1),Integer(0)],[Integer(0),Integer(2)]])
>>> a.fcp()
(x - 2) * (x - 1)
>>> [i for i in a.fcp()]
[(x - 2, 1), (x - 1, 1)]
>>> a = matrix(SR, Integer(5), (ellipsis_range(Integer(1),Ellipsis,Integer(5)**Integer(2))))
>>> a.fcp()
(x^2 - 65*x - 250) * x^3
>>> list(a.fcp())
[(x^2 - 65*x - 250, 1), (x, 3)]

Sage Live

a = matrix(SR,[[1,2],[3,4]])
a.fcp()
[i for i in a.fcp()]
a = matrix(SR,[[1,0],[0,2]])
a.fcp()
[i for i in a.fcp()]
a = matrix(SR, 5, [1..5^2])
a.fcp()
list(a.fcp())

function(*args)[source]¶

Return a matrix over a callable symbolic expression ring.

EXAMPLES:

Sage

sage: x, y = var('x,y')
sage: v = matrix([[x,y],[x*sin(y), 0]])
sage: w = v.function([x,y]); w
[       (x, y) |--> x        (x, y) |--> y]
[(x, y) |--> x*sin(y)        (x, y) |--> 0]
sage: w.parent()
Full MatrixSpace of 2 by 2 dense matrices over Callable function ring with arguments (x, y)

Python

>>> from sage.all import *
>>> x, y = var('x,y')
>>> v = matrix([[x,y],[x*sin(y), Integer(0)]])
>>> w = v.function([x,y]); w
[       (x, y) |--> x        (x, y) |--> y]
[(x, y) |--> x*sin(y)        (x, y) |--> 0]
>>> w.parent()
Full MatrixSpace of 2 by 2 dense matrices over Callable function ring with arguments (x, y)

Sage Live

x, y = var('x,y')
v = matrix([[x,y],[x*sin(y), 0]])
w = v.function([x,y]); w
w.parent()

jordan_form(subdivide=True, transformation=False)[source]¶

Return a Jordan normal form of self.

INPUT:

self – a square matrix
subdivide – boolean (default: True)
transformation – boolean (default: False)

OUTPUT:

If transformation is False, only a Jordan normal form (unique up to the ordering of the Jordan blocks) is returned. Otherwise, a pair (J, P) is returned, where J is a Jordan normal form and P is an invertible matrix such that self equals P * J * P^(-1).

If subdivide is True, the Jordan blocks in the returned matrix J are indicated by a subdivision in the sense of subdivide().

EXAMPLES:

We start with some examples of diagonalisable matrices:

Sage

sage: a,b,c,d = var('a,b,c,d')
sage: matrix([a]).jordan_form()
[a]
sage: matrix([[a, 0], [1, d]]).jordan_form(subdivide=True)
[d|0]
[-+-]
[0|a]
sage: matrix([[a, 0], [1, d]]).jordan_form(subdivide=False)
[d 0]
[0 a]
sage: matrix([[a, x, x], [0, b, x], [0, 0, c]]).jordan_form()
[c|0|0]
[-+-+-]
[0|b|0]
[-+-+-]
[0|0|a]

Python

>>> from sage.all import *
>>> a,b,c,d = var('a,b,c,d')
>>> matrix([a]).jordan_form()
[a]
>>> matrix([[a, Integer(0)], [Integer(1), d]]).jordan_form(subdivide=True)
[d|0]
[-+-]
[0|a]
>>> matrix([[a, Integer(0)], [Integer(1), d]]).jordan_form(subdivide=False)
[d 0]
[0 a]
>>> matrix([[a, x, x], [Integer(0), b, x], [Integer(0), Integer(0), c]]).jordan_form()
[c|0|0]
[-+-+-]
[0|b|0]
[-+-+-]
[0|0|a]

Sage Live

a,b,c,d = var('a,b,c,d')
matrix([a]).jordan_form()
matrix([[a, 0], [1, d]]).jordan_form(subdivide=True)
matrix([[a, 0], [1, d]]).jordan_form(subdivide=False)
matrix([[a, x, x], [0, b, x], [0, 0, c]]).jordan_form()

In the following examples, we compute Jordan forms of some non-diagonalisable matrices:

Sage

sage: matrix([[a, a], [0, a]]).jordan_form()
[a 1]
[0 a]
sage: matrix([[a, 0, b], [0, c, 0], [0, 0, a]]).jordan_form()
[c|0 0]
[-+---]
[0|a 1]
[0|0 a]

Python

>>> from sage.all import *
>>> matrix([[a, a], [Integer(0), a]]).jordan_form()
[a 1]
[0 a]
>>> matrix([[a, Integer(0), b], [Integer(0), c, Integer(0)], [Integer(0), Integer(0), a]]).jordan_form()
[c|0 0]
[-+---]
[0|a 1]
[0|0 a]

Sage Live

matrix([[a, a], [0, a]]).jordan_form()
matrix([[a, 0, b], [0, c, 0], [0, 0, a]]).jordan_form()

The following examples illustrate the transformation flag. Note that symbolic expressions may need to be simplified to make consistency checks succeed:

Sage

sage: A = matrix([[x - a*c, a^2], [-c^2, x + a*c]])
sage: J, P = A.jordan_form(transformation=True)
sage: J, P
(
[x 1]  [-a*c    1]
[0 x], [-c^2    0]
)
sage: A1 = P * J * ~P; A1
[             -a*c + x (a*c - x)*a/c + a*x/c]
[                 -c^2               a*c + x]
sage: A1.simplify_rational() == A
True

sage: B = matrix([[a, b, c], [0, a, d], [0, 0, a]])
sage: J, T = B.jordan_form(transformation=True)
sage: J, T
(
[a 1 0]  [b*d   c   0]
[0 a 1]  [  0   d   0]
[0 0 a], [  0   0   1]
)
sage: (B * T).simplify_rational() == T * J
True

Python

>>> from sage.all import *
>>> A = matrix([[x - a*c, a**Integer(2)], [-c**Integer(2), x + a*c]])
>>> J, P = A.jordan_form(transformation=True)
>>> J, P
(
[x 1]  [-a*c    1]
[0 x], [-c^2    0]
)
>>> A1 = P * J * ~P; A1
[             -a*c + x (a*c - x)*a/c + a*x/c]
[                 -c^2               a*c + x]
>>> A1.simplify_rational() == A
True

>>> B = matrix([[a, b, c], [Integer(0), a, d], [Integer(0), Integer(0), a]])
>>> J, T = B.jordan_form(transformation=True)
>>> J, T
(
[a 1 0]  [b*d   c   0]
[0 a 1]  [  0   d   0]
[0 0 a], [  0   0   1]
)
>>> (B * T).simplify_rational() == T * J
True

Sage Live

A = matrix([[x - a*c, a^2], [-c^2, x + a*c]])
J, P = A.jordan_form(transformation=True)
J, P
A1 = P * J * ~P; A1
A1.simplify_rational() == A
B = matrix([[a, b, c], [0, a, d], [0, 0, a]])
J, T = B.jordan_form(transformation=True)
J, T
(B * T).simplify_rational() == T * J

Finally, some examples involving square roots:

Sage

sage: matrix([[a, -b], [b, a]]).jordan_form()
[a - I*b|      0]
[-------+-------]
[      0|a + I*b]
sage: matrix([[a, b], [c, d]]).jordan_form(subdivide=False)
[1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2)                                                   0]
[                                                  0 1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2)]

Python

>>> from sage.all import *
>>> matrix([[a, -b], [b, a]]).jordan_form()
[a - I*b|      0]
[-------+-------]
[      0|a + I*b]
>>> matrix([[a, b], [c, d]]).jordan_form(subdivide=False)
[1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2)                                                   0]
[                                                  0 1/2*a + 1/2*d + 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2)]

Sage Live

matrix([[a, -b], [b, a]]).jordan_form()
matrix([[a, b], [c, d]]).jordan_form(subdivide=False)

minpoly(var='x')[source]¶

Return the minimal polynomial of self.

EXAMPLES:

Sage

sage: M = Matrix.identity(SR, 2)
sage: M.minpoly()
x - 1

sage: t = var('t')
sage: m = matrix(2, [1, 2, 4, t])
sage: m.minimal_polynomial()
x^2 + (-t - 1)*x + t - 8

Python

>>> from sage.all import *
>>> M = Matrix.identity(SR, Integer(2))
>>> M.minpoly()
x - 1

>>> t = var('t')
>>> m = matrix(Integer(2), [Integer(1), Integer(2), Integer(4), t])
>>> m.minimal_polynomial()
x^2 + (-t - 1)*x + t - 8

Sage Live

M = Matrix.identity(SR, 2)
M.minpoly()
t = var('t')
m = matrix(2, [1, 2, 4, t])
m.minimal_polynomial()

number_of_arguments()[source]¶

Return the number of arguments that self can take.

EXAMPLES:

Sage

sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: m = matrix([[a, (x+y)/(x+y)], [x^2, y^2+2]]); m
[      a       1]
[    x^2 y^2 + 2]
sage: m.number_of_arguments()
3

Python

>>> from sage.all import *
>>> var('a,b,c,x,y')
(a, b, c, x, y)
>>> m = matrix([[a, (x+y)/(x+y)], [x**Integer(2), y**Integer(2)+Integer(2)]]); m
[      a       1]
[    x^2 y^2 + 2]
>>> m.number_of_arguments()
3

Sage Live

var('a,b,c,x,y')
m = matrix([[a, (x+y)/(x+y)], [x^2, y^2+2]]); m
m.number_of_arguments()

simplify()[source]¶

Simplify self.

EXAMPLES:

Sage

sage: var('x,y,z')
(x, y, z)
sage: m = matrix([[z, (x+y)/(x+y)], [x^2, y^2+2]]); m
[      z       1]
[    x^2 y^2 + 2]
sage: m.simplify()
[      z       1]
[    x^2 y^2 + 2]

Python

>>> from sage.all import *
>>> var('x,y,z')
(x, y, z)
>>> m = matrix([[z, (x+y)/(x+y)], [x**Integer(2), y**Integer(2)+Integer(2)]]); m
[      z       1]
[    x^2 y^2 + 2]
>>> m.simplify()
[      z       1]
[    x^2 y^2 + 2]

Sage Live

var('x,y,z')
m = matrix([[z, (x+y)/(x+y)], [x^2, y^2+2]]); m
m.simplify()

simplify_full()[source]¶

Simplify a symbolic matrix by calling Expression.simplify_full() componentwise.

INPUT:

self – the matrix whose entries we should simplify

OUTPUT: a copy of self with all of its entries simplified

EXAMPLES:

Symbolic matrices will have their entries simplified:

Sage

sage: a,n,k = SR.var('a,n,k')
sage: f1 = sin(x)^2 + cos(x)^2
sage: f2 = sin(x/(x^2 + x))
sage: f3 = binomial(n,k)*factorial(k)*factorial(n-k)
sage: f4 = x*sin(2)/(x^a)
sage: A = matrix(SR, [[f1,f2],[f3,f4]])
sage: A.simplify_full()
[                1    sin(1/(x + 1))]
[     factorial(n) x^(-a + 1)*sin(2)]

Python

>>> from sage.all import *
>>> a,n,k = SR.var('a,n,k')
>>> f1 = sin(x)**Integer(2) + cos(x)**Integer(2)
>>> f2 = sin(x/(x**Integer(2) + x))
>>> f3 = binomial(n,k)*factorial(k)*factorial(n-k)
>>> f4 = x*sin(Integer(2))/(x**a)
>>> A = matrix(SR, [[f1,f2],[f3,f4]])
>>> A.simplify_full()
[                1    sin(1/(x + 1))]
[     factorial(n) x^(-a + 1)*sin(2)]

Sage Live

a,n,k = SR.var('a,n,k')
f1 = sin(x)^2 + cos(x)^2
f2 = sin(x/(x^2 + x))
f3 = binomial(n,k)*factorial(k)*factorial(n-k)
f4 = x*sin(2)/(x^a)
A = matrix(SR, [[f1,f2],[f3,f4]])
A.simplify_full()

simplify_rational()[source]¶

EXAMPLES:

Sage

sage: M = matrix(SR, 3, 3, range(9)) - var('t')
sage: (~M*M)[0,0]
t*(3*(2/t + (6/t + 7)/((t - 3/t - 4)*t))*(2/t + (6/t + 5)/((t - 3/t
- 4)*t))/(t - (6/t + 7)*(6/t + 5)/(t - 3/t - 4) - 12/t - 8) + 1/t +
3/((t - 3/t - 4)*t^2)) - 6*(2/t + (6/t + 5)/((t - 3/t - 4)*t))/(t -
(6/t + 7)*(6/t + 5)/(t - 3/t - 4) - 12/t - 8) - 3*(6/t + 7)*(2/t +
(6/t + 5)/((t - 3/t - 4)*t))/((t - (6/t + 7)*(6/t + 5)/(t - 3/t -
4) - 12/t - 8)*(t - 3/t - 4)) - 3/((t - 3/t - 4)*t)
sage: expand((~M*M)[0,0])
1
sage: (~M * M).simplify_rational()
[1 0 0]
[0 1 0]
[0 0 1]

Python

>>> from sage.all import *
>>> M = matrix(SR, Integer(3), Integer(3), range(Integer(9))) - var('t')
>>> (~M*M)[Integer(0),Integer(0)]
t*(3*(2/t + (6/t + 7)/((t - 3/t - 4)*t))*(2/t + (6/t + 5)/((t - 3/t
- 4)*t))/(t - (6/t + 7)*(6/t + 5)/(t - 3/t - 4) - 12/t - 8) + 1/t +
3/((t - 3/t - 4)*t^2)) - 6*(2/t + (6/t + 5)/((t - 3/t - 4)*t))/(t -
(6/t + 7)*(6/t + 5)/(t - 3/t - 4) - 12/t - 8) - 3*(6/t + 7)*(2/t +
(6/t + 5)/((t - 3/t - 4)*t))/((t - (6/t + 7)*(6/t + 5)/(t - 3/t -
4) - 12/t - 8)*(t - 3/t - 4)) - 3/((t - 3/t - 4)*t)
>>> expand((~M*M)[Integer(0),Integer(0)])
1
>>> (~M * M).simplify_rational()
[1 0 0]
[0 1 0]
[0 0 1]

Sage Live

M = matrix(SR, 3, 3, range(9)) - var('t')
(~M*M)[0,0]
expand((~M*M)[0,0])
(~M * M).simplify_rational()

simplify_trig()[source]¶

EXAMPLES:

Sage

sage: theta = var('theta')
sage: M = matrix(SR, 2, 2, [cos(theta), sin(theta), -sin(theta), cos(theta)])
sage: ~M
[1/cos(theta) - sin(theta)^2/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta)^2)                   -sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))]
[                   sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))                                          1/(sin(theta)^2/cos(theta) + cos(theta))]
sage: (~M).simplify_trig()
[ cos(theta) -sin(theta)]
[ sin(theta)  cos(theta)]

Python

>>> from sage.all import *
>>> theta = var('theta')
>>> M = matrix(SR, Integer(2), Integer(2), [cos(theta), sin(theta), -sin(theta), cos(theta)])
>>> ~M
[1/cos(theta) - sin(theta)^2/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta)^2)                   -sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))]
[                   sin(theta)/((sin(theta)^2/cos(theta) + cos(theta))*cos(theta))                                          1/(sin(theta)^2/cos(theta) + cos(theta))]
>>> (~M).simplify_trig()
[ cos(theta) -sin(theta)]
[ sin(theta)  cos(theta)]

Sage Live

theta = var('theta')
M = matrix(SR, 2, 2, [cos(theta), sin(theta), -sin(theta), cos(theta)])
~M
(~M).simplify_trig()

variables()[source]¶

Return the variables of self.

EXAMPLES:

Sage

sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: m = matrix([[x, x+2], [x^2, x^2+2]]); m
[      x   x + 2]
[    x^2 x^2 + 2]
sage: m.variables()
(x,)
sage: m = matrix([[a, b+c], [x^2, y^2+2]]); m
[      a   b + c]
[    x^2 y^2 + 2]
sage: m.variables()
(a, b, c, x, y)

Python

>>> from sage.all import *
>>> var('a,b,c,x,y')
(a, b, c, x, y)
>>> m = matrix([[x, x+Integer(2)], [x**Integer(2), x**Integer(2)+Integer(2)]]); m
[      x   x + 2]
[    x^2 x^2 + 2]
>>> m.variables()
(x,)
>>> m = matrix([[a, b+c], [x**Integer(2), y**Integer(2)+Integer(2)]]); m
[      a   b + c]
[    x^2 y^2 + 2]
>>> m.variables()
(a, b, c, x, y)

Sage Live

var('a,b,c,x,y')
m = matrix([[x, x+2], [x^2, x^2+2]]); m
m.variables()
m = matrix([[a, b+c], [x^2, y^2+2]]); m
m.variables()