In statistics, the matrix normal distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables.

Definition[edit]

The probability density function for the random matrix X (n × p) that follows the matrix normal distribution http://upload.wikimedia.org/math/c/8/1/c8176ca8b4e678d9ab698db1a901da6d.png has the form:

http://upload.wikimedia.org/math/8/5/9/8591ee2bb2ef655d5fa791186aa1853b.png

where M is n × p, U is n × n and V is p × p.

There are several ways to define the two covariance matrices. One possibility is

http://upload.wikimedia.org/math/6/9/0/6904a151348afe595bc950b9e1c2381a.png

http://upload.wikimedia.org/math/b/d/2/bd28fcac75eea641fd2be20bb0adbf44.png

where http://upload.wikimedia.org/math/4/a/8/4a8a08f09d37b73795649038408b5f33.png is a constant which depends on U and ensures appropriate power normalization.

The matrix normal is related to the multivariate normal distribution in the following way:

http://upload.wikimedia.org/math/6/9/0/69084eef5a07cc5b9e9be7921c187255.png

if and only if

http://upload.wikimedia.org/math/0/9/f/09f77b51a1178c6e039de4bf9e962c08.png

where http://upload.wikimedia.org/math/e/9/d/e9dd9013ec300ceba41484dfc2c9a876.png denotes the Kronecker product and http://upload.wikimedia.org/math/a/9/5/a95cd304d1f8ed28a02ef5d474276d2b.png denotes the vectorization of http://upload.wikimedia.org/math/e/d/4/ed453d7dfcbd94d16050e57e602a5f66.png.

Example[edit]

Let's imagine a sample of n independent p-dimensional random variables identically distributed according to a multivariate normal distribution:

http://upload.wikimedia.org/math/e/d/4/ed4ef77f7e281173593b747c7a3543a7.png.

When defining the n × p matrix http://upload.wikimedia.org/math/5/9/8/598f6444904755dda4a859a1e377468e.png for which the ith row is http://upload.wikimedia.org/math/9/3/a/93ad30d3d284e9ae549acaa2e11b0b0b.png, we obtain:

http://upload.wikimedia.org/math/7/a/c/7acd791028212972274f6c5ea4c4549b.png

where each row of http://upload.wikimedia.org/math/e/d/4/ed453d7dfcbd94d16050e57e602a5f66.png is equal to http://upload.wikimedia.org/math/3/5/1/3518f9afc5ae15868be24d59aa75aefb.png, that is http://upload.wikimedia.org/math/c/d/e/cde39ddaaa78c0006d9807fa457e0102.png is the n × n identity matrix, that is the rows are independent, and http://upload.wikimedia.org/math/9/6/e/96e376a718757cc254e4f78c7a35118d.png.

Relation to other distributions[edit]

Dawid (1981) provides a discussion of the relation of the matrix-valued normal distribution to other distributions, including the Wishart distribution, Inverse Wishart distribution and matrix t-distribution, but uses different notation from that employed here.

See also[edit]

• Multivariate normal distribution.

References[edit]

• Dawid, A.P. (1981). "Some matrix-variate distribution theory: Notational considerations and a Bayesian application". Biometrika 68 (1): 265–274. doi:10.1093/biomet/68.1.265. JSTOR 2335827. MR 614963.
• Dutilleul, P (1999). "The MLE algorithm for the matrix normal distribution". Journal of Statistical Computation and Simulation 64 (2): 105–123. doi:10.1080/00949659908811970.
• Arnold, S.F. (1981), The theory of linear models and multivariate analysis, New York: John Wiley & Sons, ISBN 0471050652