Matrices and Spaces of Matrices#

Sage provides native support for working with matrices over any commutative or noncommutative ring. The parent object for a matrix is a matrix space MatrixSpace(R, n, m) of all \(n\times m\) matrices over a ring \(R\).

To create a matrix, either use the matrix(...) function or create a matrix space using the MatrixSpace command and coerce an object into it.

Matrices also act on row vectors, which you create using the vector(...) command or by making a VectorSpace and coercing lists into it. The natural action of matrices on row vectors is from the right. Sage currently does not have a column vector class (on which matrices would act from the left), but this is planned.

In addition to native Sage matrices, Sage also includes the following additional ways to compute with matrices:

  • Several math software systems included with Sage have their own native matrix support, which can be used from Sage. E.g., PARI, GAP, Maxima, and Singular all have a notion of matrices.

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

  • The scipy module provides support for sparse numerical linear algebra, among many other things.

  • The numpy module, which you load by typing import numpy is included standard with Sage. It contains a very sophisticated and well developed array class, plus optimized support for numerical linear algebra. Sage’s matrices over RDF and CDF (native floating-point real and complex numbers) use numpy.

Finally, this module contains some data-structures for matrix-like objects like operation tables (e.g. the multiplication table of a group).

Indices and Tables#