Matrix variate probability distribution

As matrixvariate probability distributions is known in the stochastic those probability measures , on the premises of matrices are defined. They appear as distributions of random matrices .

From a measure theoretical point of view , matrix-variable probability distributions do not differ from multivariate probability distributions . This is because that the measurable amounts on the measurable quantities are identified. For the assignment of the probabilities it is therefore irrelevant whether it is a matrix with rows and columns or a vector of length . ${\displaystyle \mathbb {R} ^{n\times k}}$ ${\displaystyle \mathbb {R} ^{nk}}$ ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle nk}$

However, due to the additional algebraic structure, matrix-variable probability distributions also allow algebraic questions to be investigated with stochastic approaches. In this way questions can be investigated like

• the entries in a matrix are evenly distributed in the interval from zero to one. How are the eigenvalues distributed?
• What is the probability that the matrix is ​​invertible?

Web links

• AK Gupta: Matrix variate distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

See also

• List of multivariate and matrix-variable probability distributions