Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
•self - a square matrix over an exact field. For inexact matrices consult the numerical or symbolic
matrix classes.
•format - default: None
–’all’ - attempts to create every eigenspace. This will always be possible for matrices with
rational entries.
–’galois’ - for each irreducible factor of the characteristic polynomial, a single eigenspace will
be output for a single root/eigenvalue for the irreducible factor.
–None - Uses the ‘all’ format if the base ring is contained in an algebraically closed field which is
implemented. Otherwise, uses the ‘galois’ format.
•var - default: ‘a’ - variable name used to represent elements of the root field of each irreducible
factor of the characteristic polynomial. If var=’a’, then the root fields will be in terms of a0, a1, a2,
...., where the numbering runs across all the irreducible factors of the characteristic polynomial, even
for linear factors.
•algebraic_multiplicity - default: False - whether or not to include the algebraic multiplicity
of each eigenvalue in the output. See the discussion below.
OUTPUT:
If algebraic_multiplicity=False, return a list of pairs (e, V) where e is an eigenvalue of the matrix, and V is
the corresponding left eigenspace. For Galois conjugates of eigenvalues, there may be just one representative
eigenspace, depending on the format keyword.
If algebraic_multiplicity=True, return a list of triples (e, V, n) where e and V are as above and n is the
algebraic multiplicity of the eigenvalue.
Warning: Uses a somewhat naive algorithm (simply factors the characteristic polynomial and computes
kernels directly over the extension field).
EXAMPLES:
We compute the left eigenspaces of a 3 × 3 rational matrix. First, we request all of the eigenvalues, so the
results are in the field of algebraic numbers, QQbar. Then we request just one eigenspace per irreducible
factor of the characteristic polynomial with the galois keyword.
sage: A = matrix(QQ,3,3,range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: es = A.eigenspaces_left(format=’all’); es
[
(0, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[ 1 -2 1]),
(-1.348469228349535, Vector space of degree 3 and dimension 1 over Algebraic Field
User basis matrix:
[ 1 0.3101020514433644 -0.3797958971132713]),
(13.34846922834954, Vector space of degree 3 and dimension 1 over Algebraic Field
User basis matrix:
[ 1 1.289897948556636 1.579795897113272])
]
sage: es = A.eigenspaces_left(format=’galois’); es
[
(0, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
158 Chapter 7. Base class for matrices, part 2