Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[0.0 1.0]
[1.1 3.0]
sage: m.is_symmetric()
False
The tolerance inequality is strict: sage: m.is_symmetric(tol=0.1) False sage: m.is_symmetric(tol=0.11)
True
is_unitary(tol=1e-12, algorithm=’orthonormal’)
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.”
INPUT:
•tol - default: 1e-12 - the largest value of the absolute value of the difference between two matrix
entries for which they will still be considered equal.
•algorithm - default: ‘orthonormal’ - set to ‘orthonormal’ for a stable procedure and set to ‘naive’
for a fast procedure.
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.
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 computes the product of the conjugate-transpose with the matrix and compares
the entries to the identity matrix, 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 entries of modulus 1, which is equivalent to being unitary.
So the naive algorithm might finish fairly quickly for a matrix that is not unitary, once the product has
been computed. However, the orthonormal algorithm will compute a Schur decomposition before going
through a similar check of a matrix entry-by-entry.
EXAMPLES:
A matrix that is far from unitary.
sage: A = matrix(RDF, 4, range(16))
sage: A.conjugate().transpose()*A
[224.0 248.0 272.0 296.0]
[248.0 276.0 304.0 332.0]
[272.0 304.0 336.0 368.0]
[296.0 332.0 368.0 404.0]
sage: A.is_unitary()
False
sage: A.is_unitary(algorithm=’naive’)
False
sage: A.is_unitary(algorithm=’orthonormal’)
False
374 Chapter 19. Dense matrices using a NumPy backend.