The 'algorithm' option to `random_matrix` is very useful for producing
examples of matrices.
The algorithm='diagonalizable' option allows one to generate a
diagonalizable matrix with prescribed eigenvalues with given algebraic
multiplicities. For instance,
A = random_matrix(ZZ, 6, algorithm='diagonalizable', eigenvalues=(1, 2, 3),
dimensions=(1, 2, 3))
produces a diagonalizable matrix A with eigenvalues {1, 2, 3} with
corresponding geometric multiplicities {1, 2, 3}.
I think it would be useful to have a similar algorithm that produces a
random "nice" matrix with prescribed eigenvalues, geometric multiplicities,
and algebraic multiplicities. The syntax would be something like
A = random_matrix(ZZ, algorithm='jordanizable', eigenvalues=(1, 2, 3),
geo_mults=(1, 2, 3), alg_mults=(2, 3, 4))
This would produce a 9x9 nondiagonalizable matrix A whose eigenvalues are
{1, 2, 3} with geometric multiplicities are {1, 2, 3}.
The following code accomplishes this.
from functools import partial
from sage.all import Partitions, PositiveIntegers, ZZ, jordan_block,
matrix, random_matrix
def random_jordanizable(eigenvals, geo_mults, alg_mults=None, P_bound=5):
if alg_mults is None:
alg_mults = geo_mults
if not len(eigenvals) == len(geo_mults) == len(alg_mults):
raise ValueError('Each eval must have a geo and alg mult.')
if any(map(lambda x: x not in PositiveIntegers(), geo_mults +
alg_mults)):
raise ValueError('Each mult must be a positive integer.')
if any(map(lambda g, a: g > a, zip(geo_mults, alg_mults))):
raise ValueError('Mults must satisfy geo_mult <= alg_mult.')
n = sum(alg_mults)
J = matrix([])
for eigenval, geo_mult, alg_mult in zip(eigenvals, geo_mults,
alg_mults):
jordan_sector = partial(jordan_block, eigenval)
block_sizes = Partitions(alg_mult, length=geo_mult).random_element()
blocks = map(jordan_sector, block_sizes)
J = reduce(lambda X, Y: X.block_sum(Y), [J] + blocks)
P = random_matrix(ZZ, n, algorithm='unimodular', upper_bound=P_bound)
return P*J*P.inverse()
I'm not sure how to go about trying to incorporate this into the
random_matrix function, but this is at least a start.
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