Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: abs(abs(diff)) < 1e-14
True
TESTS:
sage: m.eigenspaces_left(algebraic_multiplicity=True)
Traceback (most recent call last):
...
ValueError: algebraic_multiplicity must be set to False for double precision matrices
left_eigenvectors()
Compute the left eigenvectors of a matrix of double precision real or complex numbers (i.e. RDF or CDF).
OUTPUT: Returns a list of triples, each of the form (e,[v],1), where e is the eigenvalue, and v is an
associated left eigenvector. If the matrix is of size n, then there are n triples. Values are computed with the
SciPy library.
The format of this output is designed to match the format for exact results. However, since matrices here
have numerical entries, the resulting eigenvalues will also be numerical. No attempt is made to determine
if two eigenvalues are equal, or if eigenvalues might actually be zero. So the algebraic multiplicity of each
eigenvalue is reported as 1. Decisions about equal eigenvalues or zero eigenvalues should be addressed in
the calling routine.
The SciPy routines used for these computations produce eigenvectors normalized to have length 1, but on
different hardware they may vary by a sign. So for doctests we have normalized output by forcing their
eigenvectors to have their first non-zero entry equal to one.
EXAMPLES:
sage: m = matrix(RDF, [[-5, 3, 2, 8],[10, 2, 4, -2],[-1, -10, -10, -17],[-2, 7, 6, 13]])
sage: m
[ -5.0 3.0 2.0 8.0]
[ 10.0 2.0 4.0 -2.0]
[ -1.0 -10.0 -10.0 -17.0]
[ -2.0 7.0 6.0 13.0]
sage: spectrum = m.left_eigenvectors()
sage: for i in range(len(spectrum)):
... spectrum[i][1][0]=matrix(RDF, spectrum[i][1]).echelon_form()[0]
sage: spectrum[0]
(2.0, [(1.0, 1.0, 1.0, 1.0)], 1)
sage: spectrum[1]
(1.0, [(1.0, 0.8, 0.8, 0.6)], 1)
sage: spectrum[2]
(-2.0, [(1.0, 0.4, 0.6, 0.2)], 1)
sage: spectrum[3]
(-1.0, [(1.0, 1.0, 2.0, 2.0)], 1)
log_determinant()
Compute the log of the absolute value of the determinant using LU decomposition.
Note: This is useful if the usual determinant overflows.
EXAMPLES:
sage: m = matrix(RDF,2,2,range(4)); m
[0.0 1.0]
[2.0 3.0]
sage: RDF(log(abs(m.determinant())))
0.69314718056
377