View source for Matrix variate distribution

A matrix random phenomenon is an observable phenomenon that can be represented in matrix form and that, under repeated observations, yields different outcomes which are not deterministically predictable. Instead, the outcomes obey certain conditions of statistical regularity. The set of descriptions of all possible outcomes that may occur on observing a matrix random phenomenon is the [[Sampling space|sampling space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m1201501.png" />. A matrix event is a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m1201502.png" />. A measure of the degree of certainty with which a given matrix event will occur when observing a matrix random phenomenon can be found by defining a [[Probability|probability]] function on subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m1201503.png" />, assigning a [[Probability|probability]] to every matrix event.

A [[Matrix|matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m1201504.png" /> consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m1201505.png" /> elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m1201506.png" /> which are real-valued functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m1201507.png" /> is a real random matrix if the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m1201508.png" /> of

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consists of Borel sets in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015010.png" />-dimensional real space and if for each [[Borel set|Borel set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015011.png" /> of real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015012.png" />-tuples, arranged in a matrix,

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in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015014.png" />, the set

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is an event in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015016.png" />. The probability density function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015017.png" /> (cf. also [[Density of a probability distribution|Density of a probability distribution]]) is a scalar function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015018.png" /> such that:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015019.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015020.png" />; and

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015022.png" /> is a subset of the space of realizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015023.png" />. A scalar function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015024.png" /> defines the joint (bi-matrix variate) probability density function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015026.png" /> if

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015027.png" />;

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015028.png" />; and

c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015030.png" /> is a subset of the space of realizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015031.png" />.

The marginal probability density function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015032.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015033.png" />, and the conditional probability density function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015034.png" /> given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015035.png" /> is defined by

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015037.png" /> is the marginal probability density function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015038.png" />.

Two random matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015040.png" /> are independently distributed if and only if

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015043.png" /> are the marginal densities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015045.png" />, respectively.

The characteristic function of the random matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015046.png" /> is defined as

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015048.png" /> is a real arbitrary matrix and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015049.png" /> is the exponential trace function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015050.png" />.

For the random matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015051.png" />, the mean matrix is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015052.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015053.png" /> covariance matrix of the random matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015055.png" /> is defined by

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==Examples of matrix variate distributions.==
The matrix variate normal distribution

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The [[Wishart distribution|Wishart distribution]]

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The matrix variate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120150/m12015062.png" />-distribution

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The matrix variate beta-type-I distribution

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The matrix variate beta-type-II distribution

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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Bougerol,   J. Lacroix,   "Products of random matrices with applications to Schrödinger operators" , Birkhäuser  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Carmeli,   "Statistical theory and random matrices" , M. Dekker  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  "Random matrices and their applications"  J.E. Cohen (ed.)  H. Kesten (ed.)  C.M. Newman (ed.) , Amer. Math. Soc.  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.K. Gupta,   T. Varga,   "Elliptically contoured models in statistics" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.K. Gupta,   V.L. Girko,   "Multidimensional statistical analysis and theory of random matrices" , VSP  (1996)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.L. Mehta,   "Random matrices" , Acad. Press  (1991)  (Edition: Second)</TD></TR></table>