Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
OUTPUT:
True if the matrix is square and equal to the transpose with every entry conjugated, and False otherwise.
Note that if conjugation has no effect on elements of the base ring (such as for integers), then the
is_symmetric() method is equivalent and faster.
The tolerance parameter is used to allow for numerical values to be equal if there is a slight difference due
to round-off and other imprecisions.
The result is cached, on a per-tolerance and per-algorithm basis.
ALGORITHMS:
The naive algorithm simply compares corresponing entries on either side of the diagonal (and on the
diagonal itself) to see if they are conjugates, with equality controlled by the tolerance parameter.
The orthonormal algorithm first computes a Schur decomposition (via the schur() method) and checks
that the result is a diagonal matrix with real entries.
So the naive algorithm can finish quickly for a matrix that is not Hermitian, while the orthonormal algorithm
will always compute a Schur decomposition before going through a similar check of the matrix
entry-by-entry.
EXAMPLES:
sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I],
... [-3 - I, -4*I, -2],
... [-1 + I, -2 - 8*I, 2 + I]])
sage: A.is_hermitian(algorithm=’orthonormal’)
False
sage: A.is_hermitian(algorithm=’naive’)
False
sage: B = A*A.conjugate_transpose()
sage: B.is_hermitian(algorithm=’orthonormal’)
True
sage: B.is_hermitian(algorithm=’naive’)
True
A matrix that is nearly Hermitian, but for one non-real diagonal entry.
sage: A = matrix(CDF, [[ 2, 2-I, 1+4*I],
... [ 2+I, 3+I, 2-6*I],
... [1-4*I, 2+6*I, 5]])
sage: A.is_hermitian(algorithm=’orthonormal’)
False
sage: A[1,1] = 132
sage: A.is_hermitian(algorithm=’orthonormal’)
True
We get a unitary matrix from the SVD routine and use this numerical matrix to create a matrix that should
be Hermitian (indeed it should be the identity matrix), but with some imprecision. We use this to illustrate
that if the tolerance is set too small, then we can be too strict about the equality of entries and may achieve
the wrong result (depending on the system):
sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I],
... [-3 - I, -4*I, -2],
... [-1 + I, -2 - 8*I, 2 + I]])
sage: U, _, _ = A.SVD()
sage: B=U*U.conjugate_transpose()
sage: B.is_hermitian(algorithm=’naive’)
True
368 Chapter 19. Dense matrices using a NumPy backend.