Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
•subdivide - default: False - determines if the returned matrix is subdivided. See the description
of the (output) below for details.
•kwds - additional keywords that can be passed to the method that computes the echelon form.
OUTPUT:
If A is an m × n matrix, add the m columns of an m × m identity matrix to the right of self. Then
row-reduce this m × (n + m) matrix. This matrix is returned as an immutable matrix.
If subdivide is True then the returned matrix has a single division among the columns and a single
division among the rows. The column subdivision has n columns to the left and m columns to the right.
The row division separates the non-zero rows from the zero rows, when restricted to the first n columns.
For a nonsingular matrix the final m columns of the extended echelon form are the inverse of self. For
a matrix of any size, the final m columns provide a matrix that transforms self to echelon form when
it multiplies self from the left. When the base ring is a field, the uniqueness of reduced row-echelon
form implies that this transformation matrix can be taken as the coefficients giving a canonical set of linear
combinations of the rows of self that yield reduced row-echelon form.
When subdivided as described above, and again over a field, the parts of the subdivision in the upper-left
corner and lower-right corner satisfy several interesting relationships with the row space, column space,
left kernel and right kernel of self. See the examples below.
Note: This method returns an echelon form. If the base ring is not a field, no atttempt is made to move to
the fraction field. See an example below where the base ring is changed manually.
EXAMPLES:
The four relationships at the end of this example hold in general.
sage: A = matrix(QQ, [[2, -1, 7, -1, 0, -3],
... [-1, 1, -5, 3, 4, 4],
... [2, -1, 7, 0, 2, -2],
... [2, 0, 4, 3, 6, 1],
... [2, -1, 7, 0, 2, -2]])
sage: E = A.extended_echelon_form(subdivide=True); E
[ 1 0 2 0 0 -1| 0 -1 0 1 -1]
[ 0 1 -3 0 -2 0| 0 -2 0 2 -3]
[ 0 0 0 1 2 1| 0 2/3 0 -1/3 2/3]
[-----------------------------+------------------------]
[ 0 0 0 0 0 0| 1 2/3 0 -1/3 -1/3]
[ 0 0 0 0 0 0| 0 0 1 0 -1]
sage: J = E.matrix_from_columns(range(6,11)); J
[ 0 -1 0 1 -1]
[ 0 -2 0 2 -3]
[ 0 2/3 0 -1/3 2/3]
[ 1 2/3 0 -1/3 -1/3]
[ 0 0 1 0 -1]
sage: J*A == A.rref()
True
sage: C = E.subdivision(0,0); C
[ 1 0 2 0 0 -1]
[ 0 1 -3 0 -2 0]
[ 0 0 0 1 2 1]
sage: L = E.subdivision(1,1); L
[ 1 2/3 0 -1/3 -1/3]
[ 0 0 1 0 -1]
sage: A.right_kernel() == C.right_kernel()
170 Chapter 7. Base class for matrices, part 2