Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: m = matrix( GF(17), 3, 3, [11,5,1,3,6,8,1,16,0]); d,u,v = m.smith_form()
sage: d
[1 0 0]
[0 1 0]
[0 0 0]
sage: u*m*v == d
True
Some examples over non-PID’s work anyway:
sage: R = EquationOrder(x^2 + 5, ’s’) # class number 2
sage: s = R.ring_generators()[0]
sage: A = matrix(R, 2, 2, [s-1,-s,-s,2*s+1])
sage: D, U, V = A.smith_form()
sage: D, U, V
(
[ 1 0] [ 4 s + 4] [ 1 -5*s + 6]
[ 0 -s - 6], [ s s - 1], [ 0 1]
)
sage: D == U*A*V
True
Others don’t, but they fail quite constructively:
sage: matrix(R,2,2,[s-1,-s-2,-2*s,-s-2]).smith_form()
Traceback (most recent call last):
...
ArithmeticError: Ideal Fractional ideal (2, s + 1) not principal
Empty matrices are handled safely:
sage: m = MatrixSpace(OE, 2,0)(0); d,u,v=m.smith_form(); u*m*v == d
True
sage: m = MatrixSpace(OE, 0,2)(0); d,u,v=m.smith_form(); u*m*v == d
True
sage: m = MatrixSpace(OE, 0,0)(0); d,u,v=m.smith_form(); u*m*v == d
True
Some pathological cases that crashed earlier versions:
sage: m = Matrix(OE, [[2*w,2*w-1,-w+1],[2*w+2,-2*w-1,w-1],[-2*w-1,-2*w-2,2*w-1]]); d, u, v =
True
sage: m = matrix(OE, 3, 3, [-5*w-1,-2*w-2,4*w-10,8*w,-w,w-1,-1,1,-8]); d,u,v = m.smith_form(
True
solve_left(B, check=True)
If self is a matrix A, then this function returns a vector or matrix X such that XA = B. If B is a vector
then X is a vector and if B is a matrix, then X is a matrix.
INPUT:
•B - a matrix
•check - bool (default: True) - if False and self is nonsquare, may not raise an error message even if
there is no solution. This is faster but more dangerous.
EXAMPLES:
sage: A = matrix(QQ,4,2, [0, -1, 1, 0, -2, 2, 1, 0])
sage: B = matrix(QQ,2,2, [1, 0, 1, -1])
sage: X = A.solve_left(B)
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