Linear Algebra¶
Sage provides standard constructions from linear algebra, e.g., the characteristic polynomial, echelon form, trace, decomposition, etc., of a matrix.
Creation of matrices and matrix multiplication is easy and natural:
sage: A = Matrix([[1,2,3],[3,2,1],[1,1,1]])
sage: w = vector([1,1,-4])
sage: w*A
(0, 0, 0)
sage: A*w
(-9, 1, -2)
sage: kernel(A)
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[ 1 1 -4]
>>> from sage.all import *
>>> A = Matrix([[Integer(1),Integer(2),Integer(3)],[Integer(3),Integer(2),Integer(1)],[Integer(1),Integer(1),Integer(1)]])
>>> w = vector([Integer(1),Integer(1),-Integer(4)])
>>> w*A
(0, 0, 0)
>>> A*w
(-9, 1, -2)
>>> kernel(A)
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[ 1 1 -4]
A = Matrix([[1,2,3],[3,2,1],[1,1,1]]) w = vector([1,1,-4]) w*A A*w kernel(A)
Note that in Sage, the kernel of a matrix \(A\) is the “left kernel”, i.e. the space of vectors \(w\) such that \(wA=0\).
Solving matrix equations is easy, using the method solve_right.
Evaluating A.solve_right(Y) returns a matrix (or vector)
\(X\) so that \(AX=Y\):
sage: Y = vector([0, -4, -1])
sage: X = A.solve_right(Y)
sage: X
(-2, 1, 0)
sage: A * X # checking our answer...
(0, -4, -1)
>>> from sage.all import *
>>> Y = vector([Integer(0), -Integer(4), -Integer(1)])
>>> X = A.solve_right(Y)
>>> X
(-2, 1, 0)
>>> A * X # checking our answer...
(0, -4, -1)
Y = vector([0, -4, -1]) X = A.solve_right(Y) X A * X # checking our answer...
If there is no solution, Sage returns an error:
sage: A.solve_right(w)
Traceback (most recent call last):
...
ValueError: matrix equation has no solutions
>>> from sage.all import *
>>> A.solve_right(w)
Traceback (most recent call last):
...
ValueError: matrix equation has no solutions
A.solve_right(w)
Similarly, use A.solve_left(Y) to solve for \(X\) in
\(XA=Y\).
Sage can also compute eigenvalues and eigenvectors:
sage: A = matrix([[0, 4], [-1, 0]])
sage: A.eigenvalues ()
[-2*I, 2*I]
sage: B = matrix([[1, 3], [3, 1]])
sage: B.eigenvectors_left()
[(4, [
(1, 1)
], 1), (-2, [
(1, -1)
], 1)]
>>> from sage.all import *
>>> A = matrix([[Integer(0), Integer(4)], [-Integer(1), Integer(0)]])
>>> A.eigenvalues ()
[-2*I, 2*I]
>>> B = matrix([[Integer(1), Integer(3)], [Integer(3), Integer(1)]])
>>> B.eigenvectors_left()
[(4, [
(1, 1)
], 1), (-2, [
(1, -1)
], 1)]
A = matrix([[0, 4], [-1, 0]]) A.eigenvalues () B = matrix([[1, 3], [3, 1]]) B.eigenvectors_left()
(The syntax for the output of eigenvectors_left is a list of
triples: (eigenvalue, eigenvector, multiplicity).) Eigenvalues and
eigenvectors over QQ or RR can also be computed
using Maxima (see Maxima below).
As noted in Basic Rings, the ring over which a matrix is
defined affects some of its properties. In the following, the first
argument to the matrix command tells Sage to view the matrix as a
matrix of integers (the ZZ case), a matrix of rational numbers
(QQ), or a matrix of reals (RR):
sage: AZ = matrix(ZZ, [[2,0], [0,1]])
sage: AQ = matrix(QQ, [[2,0], [0,1]])
sage: AR = matrix(RR, [[2,0], [0,1]])
sage: AZ.echelon_form()
[2 0]
[0 1]
sage: AQ.echelon_form()
[1 0]
[0 1]
sage: AR.echelon_form()
[ 1.00000000000000 0.000000000000000]
[0.000000000000000 1.00000000000000]
>>> from sage.all import *
>>> AZ = matrix(ZZ, [[Integer(2),Integer(0)], [Integer(0),Integer(1)]])
>>> AQ = matrix(QQ, [[Integer(2),Integer(0)], [Integer(0),Integer(1)]])
>>> AR = matrix(RR, [[Integer(2),Integer(0)], [Integer(0),Integer(1)]])
>>> AZ.echelon_form()
[2 0]
[0 1]
>>> AQ.echelon_form()
[1 0]
[0 1]
>>> AR.echelon_form()
[ 1.00000000000000 0.000000000000000]
[0.000000000000000 1.00000000000000]
AZ = matrix(ZZ, [[2,0], [0,1]]) AQ = matrix(QQ, [[2,0], [0,1]]) AR = matrix(RR, [[2,0], [0,1]]) AZ.echelon_form() AQ.echelon_form() AR.echelon_form()
For computing eigenvalues and eigenvectors of matrices over floating
point real or complex numbers, the matrix should be defined over RDF
(Real Double Field) or CDF (Complex Double Field), respectively. If no
ring is specified and floating point real or complex numbers are used then
by default the matrix is defined over the RR or CC fields,
respectively, which do not support these computations for all the cases:
sage: ARDF = matrix(RDF, [[1.2, 2], [2, 3]])
sage: ARDF.eigenvalues() # rel tol 8e-16
[-0.09317121994613098, 4.293171219946131]
sage: ACDF = matrix(CDF, [[1.2, I], [2, 3]])
sage: ACDF.eigenvectors_right() # rel tol 3e-15
[(0.8818456983293743 - 0.8209140653434135*I, [(0.7505608183809549, -0.616145932704589 + 0.2387941530333261*I)], 1),
(3.3181543016706256 + 0.8209140653434133*I, [(0.14559469829270957 + 0.3756690858502104*I, 0.9152458258662108)], 1)]
>>> from sage.all import *
>>> ARDF = matrix(RDF, [[RealNumber('1.2'), Integer(2)], [Integer(2), Integer(3)]])
>>> ARDF.eigenvalues() # rel tol 8e-16
[-0.09317121994613098, 4.293171219946131]
>>> ACDF = matrix(CDF, [[RealNumber('1.2'), I], [Integer(2), Integer(3)]])
>>> ACDF.eigenvectors_right() # rel tol 3e-15
[(0.8818456983293743 - 0.8209140653434135*I, [(0.7505608183809549, -0.616145932704589 + 0.2387941530333261*I)], 1),
(3.3181543016706256 + 0.8209140653434133*I, [(0.14559469829270957 + 0.3756690858502104*I, 0.9152458258662108)], 1)]
ARDF = matrix(RDF, [[1.2, 2], [2, 3]]) ARDF.eigenvalues() # rel tol 8e-16 ACDF = matrix(CDF, [[1.2, I], [2, 3]]) ACDF.eigenvectors_right() # rel tol 3e-15
Matrix spaces¶
We create the space \(\text{Mat}_{3\times 3}(\QQ)\) of \(3 \times 3\) matrices with rational entries:
sage: M = MatrixSpace(QQ,3)
sage: M
Full MatrixSpace of 3 by 3 dense matrices over Rational Field
>>> from sage.all import *
>>> M = MatrixSpace(QQ,Integer(3))
>>> M
Full MatrixSpace of 3 by 3 dense matrices over Rational Field
M = MatrixSpace(QQ,3) M
(To specify the space of 3 by 4 matrices, you would use
MatrixSpace(QQ,3,4). If the number of columns is omitted, it
defaults to the number of rows, so MatrixSpace(QQ,3) is a synonym
for MatrixSpace(QQ,3,3).) The space of matrices is equipped with
its canonical basis:
sage: B = M.basis()
sage: len(B)
9
sage: B[0,1]
[0 1 0]
[0 0 0]
[0 0 0]
>>> from sage.all import *
>>> B = M.basis()
>>> len(B)
9
>>> B[Integer(0),Integer(1)]
[0 1 0]
[0 0 0]
[0 0 0]
B = M.basis() len(B) B[0,1]
We create a matrix as an element of M.
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
>>> from sage.all import *
>>> A = M(range(Integer(9))); A
[0 1 2]
[3 4 5]
[6 7 8]
A = M(range(9)); A
Next we compute its reduced row echelon form and kernel.
sage: A.echelon_form()
[ 1 0 -1]
[ 0 1 2]
[ 0 0 0]
sage: A.kernel()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2 1]
>>> from sage.all import *
>>> A.echelon_form()
[ 1 0 -1]
[ 0 1 2]
[ 0 0 0]
>>> A.kernel()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2 1]
A.echelon_form() A.kernel()
Next we illustrate computation of matrices defined over finite fields:
sage: M = MatrixSpace(GF(2),4,8)
sage: A = M([1,1,0,0, 1,1,1,1, 0,1,0,0, 1,0,1,1,
....: 0,0,1,0, 1,1,0,1, 0,0,1,1, 1,1,1,0])
sage: A
[1 1 0 0 1 1 1 1]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 1 1 1 1 1 0]
sage: rows = A.rows()
sage: A.columns()
[(1, 0, 0, 0), (1, 1, 0, 0), (0, 0, 1, 1), (0, 0, 0, 1),
(1, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0)]
sage: rows
[(1, 1, 0, 0, 1, 1, 1, 1), (0, 1, 0, 0, 1, 0, 1, 1),
(0, 0, 1, 0, 1, 1, 0, 1), (0, 0, 1, 1, 1, 1, 1, 0)]
>>> from sage.all import *
>>> M = MatrixSpace(GF(Integer(2)),Integer(4),Integer(8))
>>> A = M([Integer(1),Integer(1),Integer(0),Integer(0), Integer(1),Integer(1),Integer(1),Integer(1), Integer(0),Integer(1),Integer(0),Integer(0), Integer(1),Integer(0),Integer(1),Integer(1),
... Integer(0),Integer(0),Integer(1),Integer(0), Integer(1),Integer(1),Integer(0),Integer(1), Integer(0),Integer(0),Integer(1),Integer(1), Integer(1),Integer(1),Integer(1),Integer(0)])
>>> A
[1 1 0 0 1 1 1 1]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 1 1 1 1 1 0]
>>> rows = A.rows()
>>> A.columns()
[(1, 0, 0, 0), (1, 1, 0, 0), (0, 0, 1, 1), (0, 0, 0, 1),
(1, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0)]
>>> rows
[(1, 1, 0, 0, 1, 1, 1, 1), (0, 1, 0, 0, 1, 0, 1, 1),
(0, 0, 1, 0, 1, 1, 0, 1), (0, 0, 1, 1, 1, 1, 1, 0)]
M = MatrixSpace(GF(2),4,8)
A = M([1,1,0,0, 1,1,1,1, 0,1,0,0, 1,0,1,1,
0,0,1,0, 1,1,0,1, 0,0,1,1, 1,1,1,0])
A
rows = A.rows()
A.columns()
rows
We make the subspace over \(\GF{2}\) spanned by the above rows.
sage: V = VectorSpace(GF(2),8)
sage: S = V.subspace(rows)
sage: S
Vector space of degree 8 and dimension 4 over Finite Field of size 2
Basis matrix:
[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]
sage: A.echelon_form()
[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]
>>> from sage.all import *
>>> V = VectorSpace(GF(Integer(2)),Integer(8))
>>> S = V.subspace(rows)
>>> S
Vector space of degree 8 and dimension 4 over Finite Field of size 2
Basis matrix:
[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]
>>> A.echelon_form()
[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]
V = VectorSpace(GF(2),8) S = V.subspace(rows) S A.echelon_form()
The basis of \(S\) used by Sage is obtained from the non-zero rows of the reduced row echelon form of the matrix of generators of \(S\).
Sparse Linear Algebra¶
Sage has support for sparse linear algebra over PIDs.
sage: M = MatrixSpace(QQ, 100, sparse=True)
sage: A = M.random_element(density = 0.05)
sage: E = A.echelon_form()
>>> from sage.all import *
>>> M = MatrixSpace(QQ, Integer(100), sparse=True)
>>> A = M.random_element(density = RealNumber('0.05'))
>>> E = A.echelon_form()
M = MatrixSpace(QQ, 100, sparse=True) A = M.random_element(density = 0.05) E = A.echelon_form()
The multi-modular algorithm in Sage is good for square matrices (but not so good for non-square matrices):
sage: M = MatrixSpace(QQ, 50, 100, sparse=True)
sage: A = M.random_element(density = 0.05)
sage: E = A.echelon_form()
sage: M = MatrixSpace(GF(2), 20, 40, sparse=True)
sage: A = M.random_element()
sage: E = A.echelon_form()
>>> from sage.all import *
>>> M = MatrixSpace(QQ, Integer(50), Integer(100), sparse=True)
>>> A = M.random_element(density = RealNumber('0.05'))
>>> E = A.echelon_form()
>>> M = MatrixSpace(GF(Integer(2)), Integer(20), Integer(40), sparse=True)
>>> A = M.random_element()
>>> E = A.echelon_form()
M = MatrixSpace(QQ, 50, 100, sparse=True) A = M.random_element(density = 0.05) E = A.echelon_form() M = MatrixSpace(GF(2), 20, 40, sparse=True) A = M.random_element() E = A.echelon_form()
Note that Python is case sensitive:
sage: M = MatrixSpace(QQ, 10,10, Sparse=True)
Traceback (most recent call last):
...
TypeError: ...__init__() got an unexpected keyword argument 'Sparse'
>>> from sage.all import *
>>> M = MatrixSpace(QQ, Integer(10),Integer(10), Sparse=True)
Traceback (most recent call last):
...
TypeError: ...__init__() got an unexpected keyword argument 'Sparse'
M = MatrixSpace(QQ, 10,10, Sparse=True)