.. _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]

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: MatrixSpace() got an unexpected keyword argument 'Sparse'