Alternatively, built in method random_matrix can be used, while explicitly declaring the (ring, size, and) random eigenvalues, and their dimensions. Here, the sample code uses a poisson distribution with parameter lambda=7 of the given size. (Feel free to fill in in an other way the diagonal.)
from scipy.stats import poisson
# my way to get some random eigenvalues involves the above...
# use an other way to generate diag, if this is too artificial
N = 10
# we will produce an NxN positive definite random matrix with integer eigenvalues
diag = list( poisson.rvs( 7, size=N ) )
diag = [ 1+d for d in diag ] # all entries are now > 0
evs = list( set( diag ) )
dims = [ diag.count( ev ) for ev in evs ]
print "diag =", diag
print "evs =", evs
print "dims =", dims
X = random_matrix( ZZ
, N
, algorithm ='diagonalizable'
, eigenvalues=evs
, dimensions =dims )
print X
This gives this time:
diag = [9, 6, 11, 12, 3, 3, 10, 8, 10, 5]
evs = [3, 5, 6, 8, 9, 10, 11, 12]
dims = [2, 1, 1, 1, 1, 2, 1, 1]
[ 15 8 4 2 -2 -4 0 14 66 -24]
[ 24 54 80 -24 -128 108 28 0 1012 384]
[ 44 64 96 -20 -132 100 28 28 1152 348]
[ -28 -44 -64 25 98 -77 -21 -14 -834 -272]
[ -17 -28 -42 11 76 -52 -14 -7 -549 -186]
[ 2 16 36 -14 -62 70 14 -14 452 202]
[ -2 -16 -36 14 62 -58 -4 14 -448 -206]
[ -8 8 32 -16 -60 62 14 -25 368 248]
[ -6 -12 -20 6 32 -27 -7 0 -247 -96]
[ -5 -4 -2 -1 1 2 0 -7 -33 17]
