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.

Matrix normal
Notation
Parameters

location (real matrix)
scale (positive-definite real matrix)

scale (positive-definite real matrix)
Support
PDF
Mean
Variance (among-row) and (among-column)

DefinitionEdit

The probability density function for the random matrix X (n × p) that follows the matrix normal distribution   has the form:

 

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

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

 

if and only if

 

where   denotes the Kronecker product and   denotes the vectorization of  .

ProofEdit

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:

 

which is the argument of the exponent of the multivariate normal PDF. The proof is completed by using the determinant property:  

PropertiesEdit

If  , then we have the following properties:[1][2]

Expected valuesEdit

The mean, or expected value is:

 

and we have the following second-order expectations:

 
 

where   denotes trace.

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

 

TransformationEdit

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:

 

ExampleEdit

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

 .

When defining the n × p matrix   for which the ith row is  , we obtain:

 

where each row of   is equal to  , that is  ,   is the n × n identity matrix, that is the rows are independent, and  .

Maximum likelihood parameter estimationEdit

Given k matrices, each of size n × p, denoted  , 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 [3] 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 distributionEdit

Sampling from the matrix normal distribution is a special case of the sampling procedure for the multivariate normal distribution. Let   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 distributionsEdit

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 alsoEdit

ReferencesEdit

  1. ^ A K Gupta; D K Nagar (22 October 1999). "Chapter 2: MATRIX VARIATE NORMAL DISTRIBUTION". Matrix Variate Distributions. CRC Press. ISBN 978-1-58488-046-2. Retrieved 23 May 2014.
  2. ^ Ding, Shanshan; R. Dennis Cook (2014). "DIMENSION FOLDING PCA AND PFC FOR MATRIX- VALUED PREDICTORS". Statistica Sinica. 24 (1): 463–492.
  3. ^ Glanz, Hunter; Carvalho, Luis. "An Expectation-Maximization Algorithm for the Matrix Normal Distribution". arXiv:1309.6609.