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

Definition

The probability density function for the random matrix X (np has the form:

denotes trace and M is np, U is nn and V is pp.

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

if and only if

denotes the Kronecker product denotes the vectorization.

Proof

The equivalence between the above matrix normal and multivariate normal density functions can be shown using several properties of the trace and Kronecker product, as follows. We start with the argument of the exponent of the matrix normal PDF:

Properties

, then we have the following properties:

Expected values

The mean, or expected value is:

and we have the following second-order expectations:

denotes trace.

More generally, for appropriately dimensioned matrices A,B,C:

Transformation

Transpose transform:

Linear transform: let D (r-by-n), be of full rank r ≤ n and C (p-by-s), be of full rank s ≤ p, then:

Composition

The product of two matrix normal densities

is proportional to a normal density with parameters:

Note: this is distinct from the product of normal variables, which is not generally normal.

Example

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

1. .

When defining the np for which the i, we obtain:

is the nn.

Maximum likelihood parameter estimation

Given k matrices, each of size n×p, which we assume have been sampled i.i.d. from a matrix normal distribution, the maximum likelihood estimate of the parameters can be obtained by maximizing:

The solution for the mean has a closed form, namely

but the covariance parameters do not. However, these parameters can be iteratively maximized by zero-ing their gradients at:

and

See for example and references therein. The covariance parameters are non-identifiable in the sense that for any scale factor, s>0, we have:

Drawing values from the distribution

Sampling from the matrix normal distribution is a special case of the sampling procedure for the multivariate normal distribution be an n by p matrix of np independent samples from the standard normal distribution, so that

Then let

so that

where A and B can be chosen by Cholesky decomposition or a similar matrix square root operation.

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

See also

• Multivariate normal distribution

References