Matrix gamma
Notation	${\displaystyle {\mathrm{M}\mathrm{G}}_p(\alpha ,\beta ,\mathbf{\Sigma })}$
Parameters	${\displaystyle \alpha >\frac{p-1}{2}}$shape parameter (real) ${\displaystyle \beta >0}$scale parameter ${\displaystyle \mathbf{\Sigma }}$scale (positive-definite real${\displaystyle p\times p}$matrix)
Support	${\displaystyle \mathbf{X}}$positive-definite real${\displaystyle p\times p}$matrix
PDF	${\displaystyle \frac{|\mathbf{\Sigma }{|}^{-\alpha }}{{\beta }^{p\alpha }{\mathrm{\Gamma }}_p(\alpha )}|\mathbf{X}{|}^{\alpha -\frac{p+1}{2}}\exp \left(\mathrm{t}\mathrm{r}\left(-\frac{1}{\beta }{\mathbf{\Sigma }}^{-1}\mathbf{X}\right)\right)}$ ${\displaystyle {\mathrm{\Gamma }}_p}$ is themultivariate gamma function.

Instatistics, amatrix gamma distribution is a generalization of thegamma distribution topositive-definite matrices.^[1] It is effectively a different parametrization of theWishart distribution, and is used similarly, e.g. as theconjugate prior of theprecision matrix of amultivariate normal distribution andmatrix normal distribution. Thecompound distribution resulting from compounding a matrix normal with a matrix gamma prior over the precision matrix is ageneralized matrix t-distribution.^[1]

A matrix gamma distributions is identical to aWishart distribution with${\displaystyle \beta \mathbf{\Sigma }=2V,\alpha =\frac{n}{2}.}$

Notice that the parameters${\displaystyle \beta }$ and${\displaystyle \mathbf{\Sigma }}$ are not identified; the density depends on these two parameters through the product${\displaystyle \beta \mathbf{\Sigma }}$.

• inverse matrix gamma distribution.
• matrix normal distribution.
• matrix t-distribution.
• Wishart distribution.

• Gupta, A. K.; Nagar, D. K. (1999)Matrix Variate Distributions, Chapman and Hall/CRCISBN 978-1584880462

This article aboutmatrices is astub. You can help Wikipedia byexpanding it.

• v
• t
• e

• Random matrices
• Continuous distributions
• Multivariate continuous distributions
• Matrix stubs

• Articles with short description
• Short description matches Wikidata
• All stub articles