# wishart_matrix

wishart_matrix, a Python code which produces sample matrices from the Wishart or Bartlett distributions, useful for sampling random covariance matrices.

The Wishart distribution is a probability distribution for random nonnegative-definite NxN matrices that can be used to select random covariance matrices.

The objects of the distribution are NxN matrices which are the sum of DF rank-one matrices X*X' constructed from N-vectors X, where the vectors X have zero mean and covariance SIGMA. This implies that the expected value of a Wishart matrix is DF * SIGMA.

A simplified version of the Wishart distribution assumes that SIGMA is the identity matrix. We will call this the "unit Wishart distribution".

Because any Wishart matrix W is symmetric nonnegative definite, there is an upper triangular factor T so that W = T' * T. There is a corresponding Bartlett distribution of the matrices T, so that one can alternatively sample the Bartlett distribution by sampling the Bartlett distribution for T, and then forming W.

### Licensing:

The computer code and data files made available on this web page are distributed under the MIT license.

### Languages:

wishart_matrix is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

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### Reference:

• Patrick Odell, Alan Feiveson,
A numerical procedure to generate a sample covariance matrix,
Journal of the American Statistical Association,
Volume 61, Number 313, March 1966, pages 199-203.
• Stanley Sawyer,
Wishart Distributions and Inverse-Wishart Sampling,
Washington University,
30 April 2007, 12 pages.

### Source Code:

Last revised on 31 December 2023.