I would like to create a random matrix with the constraint that the matrix must be normal, i.e. the matrix and its Hermitian conjugate must commute. I would create a random matrix "without constraints" as
RandomReal[NormalDistribution[0,s],{n,n}]+
I*RandomReal[NormalDistribution[0,s],{n,n}]
But I have no idea how I can implement the constraint, that the matrix must be normal.
n x n
matrix has2 n^2
degrees of freedom. Hermiticity providesn^2
constraints, but you still haven^2
independent real numbers. These can be distributed however you like, and you will get very different behavior of matrix ensembles depending on their distributions. In Random Matrix Theory, for example, gaussian ensembles are frequently studied, where matrix entries are independent normally distributed (complex) numbers, subject to some symmetry constraints (Hermiticity for example) - and this is a well-specified problem. $\endgroup$ – Leonid Shifrin Sep 25 '12 at 13:20w[M]
as well, then your problem is well-specified. How to generate such a matrix (if it is subject to non-trivial symmetry constraints) is another matter. Random Matrix Theory is a general field which studies such random matrix ensembles. In particular, some of its most powerful results are universality statements, which state that eigenvalue statistics may be independent of the exact form of potentialw
, but depend on the symmetries only. $\endgroup$ – Leonid Shifrin Sep 25 '12 at 14:21w
is not simple, I would look at whether or not any universality holds, because it may seriously simplify it. $\endgroup$ – Leonid Shifrin Sep 25 '12 at 14:28MultivariateStatistics
package and useWishartDistribution
. For example,RandomVariate[ WishartDistribution [ IdentityMatrix@10, 20 ], 1]
$\endgroup$ – rm -rf♦ Sep 25 '12 at 16:10