Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
The naive algorithm simply compares entries of the two possible products of the matrix with its conjugatetranspose,
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. An orthonormal diagonalization is equivalent to being normal.
So the naive algorithm can finish fairly quickly for a matrix that is not normal, once the products have
been computed. However, the orthonormal algorithm will compute a Schur decomposition before going
through a similar check of a matrix entry-by-entry.
EXAMPLES:
First over the complexes. B is Hermitian, hence normal.
sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I],
... [-3 - I, -4*I, -2],
... [-1 + I, -2 - 8*I, 2 + I]])
sage: B = A*A.conjugate_transpose()
sage: B.is_hermitian()
True
sage: B.is_normal(algorithm=’orthonormal’)
True
sage: B.is_normal(algorithm=’naive’)
True
sage: B[0,0] = I
sage: B.is_normal(algorithm=’orthonormal’)
False
sage: B.is_normal(algorithm=’naive’)
False
Now over the reals. Circulant matrices are normal.
sage: G = graphs.CirculantGraph(20, [3, 7])
sage: D = digraphs.Circuit(20)
sage: A = 3*D.adjacency_matrix() - 5*G.adjacency_matrix()
sage: A = A.change_ring(RDF)
sage: A.is_normal()
True
sage: A.is_normal(algorithm = ’naive’)
True
sage: A[19,0] = 4.0
sage: A.is_normal()
False
sage: A.is_normal(algorithm = ’naive’)
False
Skew-Hermitian matrices are normal.
sage: A = matrix(CDF, [[ 1 + I, 1 - 6*I, -1 - I],
... [-3 - I, -4*I, -2],
... [-1 + I, -2 - 8*I, 2 + I]])
sage: B = A - A.conjugate_transpose()
sage: B.is_hermitian()
False
sage: B.is_normal()
True
sage: B.is_normal(algorithm=’naive’)
True
A small matrix that does not fit into any of the usual categories of normal matrices.
370 Chapter 19. Dense matrices using a NumPy backend.