Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
A singular matrix over the field QQ.
sage: A = matrix(QQ, 4, [-1,2,-3,6,0,-1,-1,0,-1,1,-5,7,-1,6,5,2])
sage: A.is_singular()
True
sage: A.right_kernel().dimension()
1
A matrix that is not singular, i.e. nonsingular, over a field.
sage: B = matrix(QQ, 4, [1,-3,-1,-5,2,-5,-2,-7,-2,5,3,4,-1,4,2,6])
sage: B.is_singular()
False
sage: B.left_kernel().dimension()
0
For “rectangular” matrices, invertibility is always False, but asking about singularity will give an error.
sage: C = matrix(QQ, 5, range(30))
sage: C.is_invertible()
False
sage: C.is_singular()
Traceback (most recent call last):
...
ValueError: self must be a square matrix
When the base ring is not a field, then a matrix may be both not invertible and not singular.
sage: D = matrix(ZZ, 4, [2,0,-4,8,2,1,-2,7,2,5,7,0,0,1,4,-6])
sage: D.is_invertible()
False
sage: D.is_singular()
False
sage: d = D.determinant(); d
2
sage: d.is_unit()
False
is_skew_symmetric()
Return True if self is a skew-symmetric matrix.
Here, “skew-symmetric matrix” means a square matrix A satisfying A T = −A. It does not require that
the diagonal entries of A are 0 (although this automatically follows from A T = −A when 2 is invertible in
the ground ring over which the matrix is considered). Skew-symmetric matrices A whose diagonal entries
are 0 are said to be “alternating”, and this property is checked by the is_alternating() method.
EXAMPLES:
sage: m = matrix(QQ, [[0,2], [-2,0]])
sage: m.is_skew_symmetric()
True
sage: m = matrix(QQ, [[1,2], [2,1]])
sage: m.is_skew_symmetric()
False
Skew-symmetric is not the same as alternating when 2 is a zero-divisor in the ground ring:
sage: n = matrix(Zmod(4), [[0, 1], [-1, 2]])
sage: n.is_skew_symmetric()
True
76 Chapter 5. Base class for matrices, part 0