Matrix Normal Distribution - Definition

Definition

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

\frac{\exp\left( -\frac{1}{2} \mbox{tr}\left \right)}{(2\pi)^{np/2} |{\boldsymbol \Omega}|^{n/2} |{\boldsymbol \Sigma}|^{p/2}}

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

{\boldsymbol \Sigma} = E\;,\;
{\boldsymbol \Omega} = E/c

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

\mathrm{vec}\;\mathbf{X} \sim N_{np}(\mathrm{vec}\;\mathbf{M}, {\boldsymbol \Omega}\otimes{\boldsymbol \Sigma}),

where denotes the Kronecker product and denotes the vectorization of .

Read more about this topic:  Matrix Normal Distribution

Famous quotes containing the word definition:

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)