Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
EXAMPLES:
sage: A = matrix(QQbar, [[ 1 + I, 1 - 6*I, -1 - I],
... [-3 - I, -4*I, -2],
... [-1 + I, -2 - 8*I, 2 + I]])
sage: A.is_hermitian()
False
sage: B = A*A.conjugate_transpose()
sage: B.is_hermitian()
True
Sage has several fields besides the entire complex numbers where conjugation is non-trivial.
sage: F. = QuadraticField(-7)
sage: C = matrix(F, [[-2*b - 3, 7*b - 6, -b + 3],
... [-2*b - 3, -3*b + 2, -2*b],
... [ b + 1, 0, -2]])
sage: C.is_hermitian()
False
sage: C = C*C.conjugate_transpose()
sage: C.is_hermitian()
True
A matrix that is nearly Hermitian, but for a non-real diagonal entry.
sage: A = matrix(QQbar, [[ 2, 2-I, 1+4*I],
... [ 2+I, 3+I, 2-6*I],
... [1-4*I, 2+6*I, 5]])
sage: A.is_hermitian()
False
sage: A[1,1] = 132
sage: A.is_hermitian()
True
Rectangular matrices are never Hermitian.
sage: A = matrix(QQbar, 3, 4)
sage: A.is_hermitian()
False
A square, empty matrix is trivially Hermitian.
sage: A = matrix(QQ, 0, 0)
sage: A.is_hermitian()
True
is_immutable()
Return True if this matrix is immutable.
See the documentation for self.set_immutable for more details about mutability.
EXAMPLES:
sage: A = Matrix(QQ[’t’,’s’], 2, 2, range(4))
sage: A.is_immutable()
False
sage: A.set_immutable()
sage: A.is_immutable()
True
is_invertible()
Return True if this matrix is invertible.
74 Chapter 5. Base class for matrices, part 0