.. _section-linalg:
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]
Note that in Sage, the kernel of a matrix :math:`A` is the
"left kernel", i.e. the space of vectors :math:`w` such that
:math:`wA=0`.
Solving matrix equations is easy, using the method ``solve_right``.
Evaluating ``A.solve_right(Y)`` returns a matrix (or vector)
:math:`X` so that :math:`AX=Y`:
.. link
::
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)
A backslash ``\`` can be used in the place of ``solve_right``; use
``A \ Y`` instead of ``A.solve_right(Y)``.
.. link
::
sage: A \ Y
(-2, 1, 0)
If there is no solution, Sage returns an error:
.. skip
::
sage: A.solve_right(w)
Traceback (most recent call last):
...
ValueError: matrix equation has no solutions
Similarly, use ``A.solve_left(Y)`` to solve for :math:`X` in
:math:`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)]
(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 :ref:`section-maxima` below).
As noted in :ref:`section-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]
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)]
Matrix spaces
-------------
We create the space :math:`\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
(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 has a basis which
Sage stores as a list:
.. link
::
sage: B = M.basis()
sage: len(B)
9
sage: B[1]
[0 1 0]
[0 0 0]
[0 0 0]
We create a matrix as an element of ``M``.
.. link
::
sage: A = M(range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
Next we compute its reduced row echelon form and kernel.
.. link
::
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]
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)]
We make the subspace over `\GF{2}` spanned by the above
rows.
.. link
::
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]
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()
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()
Note that Python is case sensitive:
::
sage: M = MatrixSpace(QQ, 10,10, Sparse=True)
Traceback (most recent call last):
...
TypeError: __classcall__() got an unexpected keyword argument 'Sparse'