Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
The matrix must have entries from a field, and it must be an exact field.
sage: A = matrix(ZZ, 4, range(16))
sage: A.is_diagonalizable()
Traceback (most recent call last):
...
ValueError: matrix entries must be from a field, not Integer Ring
sage: A = matrix(RDF, 4, range(16))
sage: A.is_diagonalizable()
Traceback (most recent call last):
...
ValueError: base field must be exact, not Real Double Field
AUTHOR:
•Rob Beezer (2011-04-01)
is_normal()
Returns True if the matrix commutes with its conjugate-transpose.
OUTPUT:
True if the matrix is square and commutes with its conjugate-transpose, and False otherwise.
Normal matrices are precisely those that can be diagonalized by a unitary matrix.
This routine is for matrices over exact rings and so may not work properly for matrices
over RR or CC. For matrices with approximate entries, the rings of doubleprecision
floating-point numbers, RDF and CDF, are a better choice since the
sage.matrix.matrix_double_dense.Matrix_double_dense.is_normal() method
has a tolerance parameter. This provides control over allowing for minor discrepancies between entries
when checking equality.
The result is cached.
EXAMPLES:
Hermitian matrices are normal.
sage: A = matrix(QQ, 5, range(25)) + I*matrix(QQ, 5, range(0, 50, 2))
sage: B = A*A.conjugate_transpose()
sage: B.is_hermitian()
True
sage: B.is_normal()
True
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.is_normal()
True
Skew-symmetric matrices are normal.
sage: A = matrix(QQ, 5, range(25))
sage: B = A - A.transpose()
sage: B.is_skew_symmetric()
True
sage: B.is_normal()
True
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