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Sage Reference Manual

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Matrices and Spaces of Matrices

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===============================

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Sage provides native support for working with matrices over any

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commutative or noncommutative ring. The parent object for a matrix

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is a matrix space ``MatrixSpace(R, n, m)`` of all

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:math:`n\times

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m` matrices over a ring :math:`R`.

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To create a matrix, either use the ``matrix(...)``

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function or create a matrix space using the

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``MatrixSpace`` command and coerce an object into it.

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Matrices also act on row vectors, which you create using the

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``vector(...)`` command or by making a

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``VectorSpace`` and coercing lists into it. The natural

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action of matrices on row vectors is from the right. Sage currently

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does not have a column vector class (on which matrices would act

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from the left), but this is planned.

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In addition to native Sage matrices, Sage also includes the

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following additional ways to compute with matrices:

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-  Several math software systems included with Sage have their own

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   native matrix support, which can be used from Sage. E.g., PARI,

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   GAP, Maxima, and Singular all have a notion of matrices.

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-  The GSL C-library is included with Sage, and can be used via

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   Cython.

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-  The ``scipy`` module provides support for

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   *sparse* numerical linear algebra, among many other things.

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-  The ``numpy`` module, which you load by typing

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   ``import numpy`` is included standard with Sage. It

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   contains a very sophisticated and well developed array class, plus

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   optimized support for *numerical linear algebra*.  Sage's matrices

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   over RDF and CDF (native floating-point real and complex numbers)

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   use numpy.

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Finally, this module contains some data-structures for matrix-like

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objects like operation tables (e.g. the multiplication table of a group).

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.. toctree::

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   :maxdepth: 2

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   sage/matrix/matrix_space

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   sage/matrix/constructor

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   sage/matrix/docs

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   sage/matrix/matrix_misc

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   sage/matrix/matrix

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   sage/matrix/matrix0

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   sage/matrix/matrix1

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   sage/matrix/matrix2

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   sage/matrix/strassen

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   sage/matrix/berlekamp_massey

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   sage/matrix/matrix_dense

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   sage/matrix/matrix_sparse

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   sage/matrix/matrix_generic_dense

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   sage/matrix/matrix_generic_sparse

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   sage/matrix/matrix_modn_sparse

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   sage/matrix/matrix_symbolic_dense

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   sage/matrix/matrix_integer_dense

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   sage/matrix/matrix_rational_dense

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   sage/matrix/matrix_double_dense

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   sage/matrix/matrix_real_double_dense

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   sage/matrix/matrix_complex_double_dense

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   sage/matrix/matrix_mpolynomial_dense

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   sage/matrix/operation_table

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   sage/matrix/action

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   sage/matrix/change_ring

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   sage/matrix/echelon_matrix

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   sage/matrix/matrix_cyclo_dense

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   sage/matrix/matrix_integer_2x2

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   sage/matrix/matrix_integer_dense_hnf

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   sage/matrix/matrix_integer_dense_saturation

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   sage/matrix/matrix_integer_sparse

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   sage/matrix/matrix_mod2_dense

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   sage/matrix/matrix_mod2e_dense

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   sage/matrix/matrix_modn_dense_double

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   sage/matrix/matrix_modn_dense_float

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   sage/matrix/matrix_rational_sparse

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   sage/matrix/matrix_window

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   sage/matrix/misc

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   sage/matrix/symplectic_basis

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   sage/matrix/benchmark

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.. include:: ../footer.txt

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