Notebox	Probability density function Probability density function In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the... :

The matrix normal distribution is a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 that is a generalization of the normal distribution to matrix-valued random variables.

Definition

The probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 for the random matrix X (n × p) that follows the matrix normal distribution has the form:
 
where M is n × p, Ω is p × p and Σ is n × n. There are several ways to define the two covariance matrices. One possibility is
where c is a constant which depends on Σ and ensures appropriate power normalization.

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

if and only if


where denotes the Kronecker product

Kronecker product

In mathematics, the Kronecker product, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It gives the matrix of the tensor product with respect to a standard choice of basis. The Kronecker product should not be confused with the usual matrix...

 and denotes the vectorization

Vectorization (mathematics)

In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector...

 of .

Example

Matrix Normal random variables arise from a sample identically distributed multivariate Normal random variables with possible dependence between the vectors. For example, if is an (n × p) matrix whose rows are independent with distribution then , where is the (n × n) identity matrix

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

. On the other hand, the columns of are dependent but identically distributed multivariate Normal random variables. Furthermore, , where .

Relation to other distributions

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 a "matrix-t distribution", but uses different notation from that employed here.