Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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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
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 True sage: A.row_space() == C.row_space() True sage: A.column_space() == L.right_kernel() True sage: A.left_kernel() == L.row_space() True For a nonsingular matrix, the right half of the extended echelon form is the inverse matrix. sage: B = matrix(QQ, [[1,3,4], [1,4,4], [0,-2,-1]]) sage: E = B.extended_echelon_form() sage: J = E.matrix_from_columns(range(3,6)); J [-4 5 4] [-1 1 0] [ 2 -2 -1] sage: J == B.inverse() True The result is in echelon form, so if the base ring is not a field, the leading entry of each row may not be 1. But you can easily change to the fraction field if necessary. sage: A = matrix(ZZ, [[16, 20, 4, 5, -4, 13, 5], ... [10, 13, 3, -3, 7, 11, 6], ... [-12, -15, -3, -3, 2, -10, -4], ... [10, 13, 3, 3, -1, 9, 4], ... [4, 5, 1, 8, -10, 1, -1]]) sage: E = A.extended_echelon_form(subdivide=True); E [ 2 0 -2 2 -9 -3 -4| 0 4 -3 -9 4] [ 0 1 1 2 0 1 1| 0 1 2 1 1] [ 0 0 0 3 -4 -1 -1| 0 3 1 -3 3] [--------------------+--------------] [ 0 0 0 0 0 0 0| 1 6 3 -6 5] [ 0 0 0 0 0 0 0| 0 7 2 -7 6] sage: J = E.matrix_from_columns(range(7,12)); J [ 0 4 -3 -9 4] [ 0 1 2 1 1] [ 0 3 1 -3 3] [ 1 6 3 -6 5] [ 0 7 2 -7 6] sage: J*A == A.echelon_form() True sage: B = A.change_ring(QQ) sage: B.extended_echelon_form(subdivide=True) [ 1 0 -1 0 -19/6 -7/6 -5/3| 0 0 -89/42 -5/2 1/7] [ 0 1 1 0 8/3 5/3 5/3| 0 0 34/21 2 -1/7] [ 0 0 0 1 -4/3 -1/3 -1/3| 0 0 1/21 0 1/7] [------------------------------------------------+----------------------------------] [ 0 0 0 0 0 0 0| 1 0 9/7 0 -1/7] [ 0 0 0 0 0 0 0| 0 1 2/7 -1 6/7] Subdivided, or not, the result is immutable, so make a copy if you want to make changes. sage: A = matrix(FiniteField(7), [[2,0,3], [5,5,3], [5,6,5]]) sage: E = A.extended_echelon_form() sage: E.is_mutable() False sage: F = A.extended_echelon_form(subdivide=True) sage: F 171
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