Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

CHAPTER
THREE
MATRICES OVER AN ARBITRARY RING
AUTHORS:
• William Stein
• Martin Albrecht: conversion to Pyrex
• Jaap Spies: various functions
• Gary Zablackis: fixed a sign bug in generic determinant.
• William Stein and Robert Bradshaw - complete restructuring.
• Rob Beezer - refactor kernel functions.
Elements of matrix spaces are of class Matrix (or a class derived from Matrix). They can be either sparse or dense,
and can be defined over any base ring.
EXAMPLES:
We create the 2 × 3 matrix
as an element of a matrix space over Q:
( 1 2
) 3
4 5 6
sage: M = MatrixSpace(QQ,2,3)
sage: A = M([1,2,3, 4,5,6]); A
[1 2 3]
[4 5 6]
sage: A.parent()
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
Alternatively, we could create A more directly as follows (which would completely avoid having to create the matrix
space):
sage: A = matrix(QQ, 2, [1,2,3, 4,5,6]); A
[1 2 3]
[4 5 6]
We next change the top-right entry of A. Note that matrix indexing is 0-based in Sage, so the top right entry is (0, 2),
which should be thought of as “row number 0, column number 2”.
sage: A[0,2] = 389
sage: A
[ 1 2 389]
[ 4 5 6]
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