Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
TypeError: similarity requires a matrix as an argument, not garbage
sage: B = copy(A)
sage: A.is_similar(B, transformation=’junk’)
Traceback (most recent call last):
...
ValueError: transformation keyword must be True or False, not junk
is_unitary()
Returns True if the columns of the matrix are an orthonormal basis.
For a matrix with real entries this determines if a matrix is “orthogonal” and for a matrix with complex
entries this determines if the matrix is “unitary.”
OUTPUT:
True if the matrix is square and its conjugate-transpose is its inverse, and False otherwise. In other
words, a matrix is orthogonal or unitary if the product of its conjugate-transpose times the matrix is the
identity matrix.
For numerical matrices a specialized routine available over RDF and CDF is a good choice.
EXAMPLES:
sage: A = matrix(QQbar, [[(1/sqrt(5))*(1+i), (1/sqrt(55))*(3+2*I), (1/sqrt(22))*(2+2*I)],
... [(1/sqrt(5))*(1-i), (1/sqrt(55))*(2+2*I), (1/sqrt(22))*(-3+I)],
... [ (1/sqrt(5))*I, (1/sqrt(55))*(3-5*I), (1/sqrt(22))*(-2)]])
sage: A.is_unitary()
True
A permutation matrix is always orthogonal.
sage: sigma = Permutation([1,3,4,5,2])
sage: P = sigma.to_matrix(); P
[1 0 0 0 0]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
sage: P.is_unitary()
True
sage: P.change_ring(GF(3)).is_unitary()
True
sage: P.change_ring(GF(3)).is_unitary()
True
A square matrix far from unitary.
sage: A = matrix(QQ, 4, range(16))
sage: A.is_unitary()
False
Rectangular matrices are never unitary.
sage: A = matrix(QQbar, 3, 4)
sage: A.is_unitary()
False
jordan_form(base_ring=None, sparse=False, subdivide=True, transformation=False, eigenvalues=None,
check_input=True)
Compute the Jordan normal form of this square matrix A, if it exists.
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