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 symplectic_form() Find a symplectic form for self if self is an anti-symmetric, alternating matrix defined over a field. Returns a pair (F, C) such that the rows of C form a symplectic basis for self and F = C * self * C.transpose(). Raises a ValueError if not over a field, or self is not anti-symmetric, or self is not alternating. Anti-symmetric means that M = −M t . Alternating means that the diagonal of M is identically zero. A symplectic basis is a basis of the form e 1 , . . . , e j , f 1 , . . . f j , z 1 , . . . , z k such that •z i Mv t = 0 for all vectors v •e i Me j t = 0 for all i, j •f i Mf j t = 0 for all i, j •e i Mf i t = 1 for all i •e i Mf j t = 0 for all i not equal j. See the example for a pictorial description of such a basis. EXAMPLES: sage: E = matrix(QQ, 8, 8, [0, -1/2, -2, 1/2, 2, 0, -2, 1, 1/2, 0, -1, -3, 0, 2, 5/2, -3, 2, [ 0 -1/2 -2 1/2 2 0 -2 1] [ 1/2 0 -1 -3 0 2 5/2 -3] [ 2 1 0 3/2 -1 0 -1 -2] [-1/2 3 -3/2 0 1 3/2 -1/2 -1/2] [ -2 0 1 -1 0 0 1 -1] [ 0 -2 0 -3/2 0 0 1/2 -2] [ 2 -5/2 1 1/2 -1 -1/2 0 -1] [ -1 3 2 1/2 1 2 1 0] sage: F, C = E.symplectic_form(); F [ 0 0 0 0 1 0 0 0] [ 0 0 0 0 0 1 0 0] [ 0 0 0 0 0 0 1 0] [ 0 0 0 0 0 0 0 1] [-1 0 0 0 0 0 0 0] [ 0 -1 0 0 0 0 0 0] [ 0 0 -1 0 0 0 0 0] [ 0 0 0 -1 0 0 0 0] sage: F == C * E * C.transpose() True tensor_product(A, subdivide=True) Returns the tensor product of two matrices. INPUT: •A - a matrix •subdivide - default: True - whether or not to return natural subdivisions with the matrix OUTPUT: Replace each element of self by a copy of A, but first create a scalar multiple of A by the element it replaces. So if self is an m × n matrix and A is a p × q matrix, then the tensor product is an mp × nq matrix. By default, the matrix will be subdivided into submatrices of size p × q. EXAMPLES: 260 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: M1=Matrix(QQ,[[-1,0],[-1/2,-1]]) sage: M2=Matrix(ZZ,[[1,-1,2],[-2,4,8]]) sage: M1.tensor_product(M2) [ -1 1 -2| 0 0 0] [ 2 -4 -8| 0 0 0] [--------------+--------------] [-1/2 1/2 -1| -1 1 -2] [ 1 -2 -4| 2 -4 -8] sage: M2.tensor_product(M1) [ -1 0| 1 0| -2 0] [-1/2 -1| 1/2 1| -1 -2] [---------+---------+---------] [ 2 0| -4 0| -8 0] [ 1 2| -2 -4| -4 -8] Subdivisions can be optionally suppressed. sage: M1.tensor_product(M2, subdivide=False) [ -1 1 -2 0 0 0] [ 2 -4 -8 0 0 0] [-1/2 1/2 -1 -1 1 -2] [ 1 -2 -4 2 -4 -8] Different base rings are handled sensibly. sage: A = matrix(ZZ, 2, 3, range(6)) sage: B = matrix(FiniteField(23), 3, 4, range(12)) sage: C = matrix(FiniteField(29), 4, 5, range(20)) sage: D = A.tensor_product(B) sage: D.parent() Full MatrixSpace of 6 by 12 dense matrices over Finite Field of size 23 sage: E = C.tensor_product(B) Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for ’*’: ’Finite Field of size 29’ and ’Full Matrix The input is checked to be sure it is a matrix. sage: A = matrix(QQ, 2, range(4)) sage: A.tensor_product(’junk’) Traceback (most recent call last): ... TypeError: tensor product requires a second matrix, not junk trace() Return the trace of self, which is the sum of the diagonal entries of self. INPUT: •self - a square matrix OUTPUT: element of the base ring of self EXAMPLES: sage: a = matrix(3,range(9)); a [0 1 2] [3 4 5] [6 7 8] sage: a.trace() 12 261
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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