Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: spectrum[0][0]
2.0
sage: (RDF^4).subspace(spectrum[0][1].basis())
Vector space of degree 4 and dimension 1 over Real Double Field
Basis matrix:
[ 1.0 -2.0 3.0 1.0]
sage: e, V = spectrum[2]
sage: v = V.basis()[0]
sage: diff = (m*v).change_ring(CDF) - e*v
sage: abs(abs(diff)) < 3e-14
True
TESTS:
sage: m.eigenspaces_right(algebraic_multiplicity=True)
Traceback (most recent call last):
...
ValueError: algebraic_multiplicity must be set to False for double precision matrices
eigenvalues(algorithm=’default’, tol=None)
Returns a list of eigenvalues.
INPUT:
•self - a square matrix
•algorithm - default: ’default’
–’default’ - applicable to any matrix with double-precision floating point entries. Uses the
eigvals() method from SciPy.
–’symmetric’ - converts the matrix into a real matrix (i.e. with entries from RDF), then applies
the algorithm for Hermitian matrices. This algorithm can be significantly faster than the
’default’ algorithm.
–’hermitian’ - uses the eigh() method from SciPy, which applies only to real symmetric
or complex Hermitian matrices. Since Hermitian is defined as a matrix equaling its conjugatetranspose,
for a matrix with real entries this property is equivalent to being symmetric. This
algorithm can be significantly faster than the ’default’ algorithm.
•’tol’ - default: None - if set to a value other than None this is interpreted as a small real number
used to aid in grouping eigenvalues that are numerically similar. See the output description for more
information.
Warning: When using the ’symmetric’ or ’hermitian’ algorithms, no check is made on the
input matrix, and only the entries below, and on, the main diagonal are employed in the computation.
Methods such as is_symmetric() and is_hermitian() could be used to verify this beforehand.
OUTPUT:
Default output for a square matrix of size n is a list of n eigenvalues from the complex double field, CDF.
If the ’symmetric’ or ’hermitian’ algorithms are chosen, the returned eigenvalues are from the
real double field, RDF.
If a tolerance is specified, an attempt is made to group eigenvalues that are numerically similar. The return
is then a list of pairs, where each pair is an eigenvalue followed by its multiplicity. The eigenvalue reported
is the mean of the eigenvalues computed, and these eigenvalues are contained in an interval (or disk) whose
radius is less than 5*tol for n < 10, 000 in the worst case.
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