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:

    It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.”
    Jane Adams (20th century)

    Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.
    Willard Van Orman Quine (b. 1908)

    The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.
    William James (1842–1910)