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Matrixt
Notation	${\displaystyle {\mathrm{T}}_{n,p}(\nu ,\mathbf{M},\mathbf{\Sigma },\mathbf{\Omega })}$
Parameters	${\displaystyle \mathbf{M}}$location (real${\displaystyle n\times p}$matrix) ${\displaystyle \mathbf{\Omega }}$scale (positive-definitereal${\displaystyle n\times n}$matrix) ${\displaystyle \mathbf{\Sigma }}$scale (positive-definite real${\displaystyle p\times p}$matrix) ${\displaystyle \nu >0}$degrees of freedom (real)
Support	${\displaystyle \mathbf{X}\in {\mathbb{R}}^{n\times p}}$
PDF	${\displaystyle \frac{{\mathrm{\Gamma }}_p\left(\frac{\nu +n+p-1}{2}\right)}{(\pi {)}^{\frac{np}{2}}{\mathrm{\Gamma }}_p\left(\frac{\nu +p-1}{2}\right)}|\mathbf{\Omega }{|}^{-\frac{n}{2}}|\mathbf{\Sigma }{|}^{-\frac{p}{2}}}$ ${\displaystyle \times {\left|{\mathbf{I}}_p+{\mathbf{\Sigma }}^{-1}(\mathbf{X}-\mathbf{M}){\mathbf{\Omega }}^{-1}(\mathbf{X}-\mathbf{M}{)}^{\mathrm{T}}\right|}^{-\frac{\nu +n+p-1}{2}}}$
CDF	No analytic expression
Mean	${\displaystyle \mathbf{M}}$ if${\displaystyle \nu >1}$, else undefined
Mode	${\displaystyle \mathbf{M}}$
Variance	${\displaystyle \mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{v}\mathrm{e}\mathrm{c}(\mathbf{X}))=\frac{\mathbf{\Sigma }\otimes \mathbf{\Omega }}{\nu -2}}$ if${\displaystyle \nu >2}$, else undefined
CF	see below

Instatistics, thematrixt-distribution (ormatrix variatet-distribution) is the generalization of themultivariatet-distribution from vectors tomatrices.^[1][2]

The matrixt-distribution shares the same relationship with the multivariatet-distribution that thematrix normal distribution shares with themultivariate normal distribution: If the matrix has only one row, or only one column, the distributions become equivalent to the corresponding (vector-)multivariate distribution. The matrixt-distribution is thecompound distribution that results from an infinitemixture of a matrix normal distribution with aninverse Wishart distribution placed over either of its covariance matrices,^[1] and the multivariatet-distribution can be generated in a similar way.^[2]

In aBayesian analysis of amultivariate linear regression model based on the matrix normal distribution, the matrixt-distribution is theposterior predictive distribution.^[3]

For a matrixt-distribution, theprobability density function at the point${\displaystyle \mathbf{X}}$ of an${\displaystyle n\times p}$ space is

${\displaystyle f(\mathbf{X};\nu ,\mathbf{M},\mathbf{\Sigma },\mathbf{\Omega })=K\times {\left|{\mathbf{I}}_n+{\mathbf{\Sigma }}^{-1}(\mathbf{X}-\mathbf{M}){\mathbf{\Omega }}^{-1}(\mathbf{X}-\mathbf{M}{)}^{\mathrm{T}}\right|}^{-\frac{\nu +n+p-1}{2}},}$

where the constant of integrationK is given by

${\displaystyle K=\frac{{\mathrm{\Gamma }}_p\left(\frac{\nu +n+p-1}{2}\right)}{(\pi {)}^{\frac{np}{2}}{\mathrm{\Gamma }}_p\left(\frac{\nu +p-1}{2}\right)}|\mathbf{\Omega }{|}^{-\frac{n}{2}}|\mathbf{\Sigma }{|}^{-\frac{p}{2}}.}$

Here${\displaystyle {\mathrm{\Gamma }}_p}$ is themultivariate gamma function.

If${\displaystyle \mathbf{X}\sim {\mathcal{T}}_{n\times p}(\nu ,\mathbf{M},\mathbf{\Sigma },\mathbf{\Omega })}$, then we have the following properties:^[2]

The mean, orexpected value is, if${\displaystyle \nu >1}$:

${\displaystyle E[\mathbf{X}]=\mathbf{M}}$

and we have the following second-order expectations, if${\displaystyle \nu >2}$:

${\displaystyle E[(\mathbf{X}-\mathbf{M})(\mathbf{X}-\mathbf{M}{)}^T]=\frac{\mathbf{\Sigma }\mathrm{tr}(\mathbf{\Omega })}{\nu -2}}$

${\displaystyle E[(\mathbf{X}-\mathbf{M}{)}^T(\mathbf{X}-\mathbf{M})]=\frac{\mathbf{\Omega }\mathrm{tr}(\mathbf{\Sigma })}{\nu -2}}$

where${\displaystyle \mathrm{tr}}$ denotestrace.

More generally, for appropriately dimensioned matricesA,B,C:

Transpose transform:

${\displaystyle {\mathbf{X}}^T\sim {\mathcal{T}}_{p\times n}(\nu ,{\mathbf{M}}^T,\mathbf{\Omega },\mathbf{\Sigma })}$

Linear transform: letA (r-by-n), be of fullrankr ≤ n andB (p-by-s), be of full ranks ≤ p, then:

${\displaystyle \mathbf{A}\mathbf{X}\mathbf{B}\sim {\mathcal{T}}_{r\times s}(\nu ,\mathbf{A}\mathbf{M}\mathbf{B},{\mathbf{A}\mathbf{\Sigma }\mathbf{A}}^T,{\mathbf{B}}^T\mathbf{\Omega }\mathbf{B})}$

Thecharacteristic function and various other properties can be derived from the re-parameterised formulation (see below).

Re-parameterized matrix t
Notation	${\displaystyle {\mathrm{T}}_{n,p}(\alpha ,\beta ,\mathbf{M},\mathbf{\Sigma },\mathbf{\Omega })}$
Parameters	${\displaystyle \mathbf{M}}$location (real${\displaystyle n\times p}$matrix) ${\displaystyle \mathbf{\Omega }}$scale (positive-definitereal${\displaystyle p\times p}$matrix) ${\displaystyle \mathbf{\Sigma }}$scale (positive-definitereal${\displaystyle n\times n}$matrix) ${\displaystyle \alpha >(p-1)/2}$shape parameter ${\displaystyle \beta >0}$scale parameter
Support	${\displaystyle \mathbf{X}\in {\mathbb{R}}^{n\times p}}$
PDF	${\displaystyle \frac{{\mathrm{\Gamma }}_p(\alpha +n/2)}{(2\pi /\beta {)}^{\frac{np}{2}}{\mathrm{\Gamma }}_p(\alpha )}|\mathbf{\Omega }{|}^{-\frac{n}{2}}|\mathbf{\Sigma }{|}^{-\frac{p}{2}}}$ ${\displaystyle \times {\left|{\mathbf{I}}_n+\frac{\beta }{2}{\mathbf{\Sigma }}^{-1}(\mathbf{X}-\mathbf{M}){\mathbf{\Omega }}^{-1}(\mathbf{X}-\mathbf{M}{)}^{\mathrm{T}}\right|}^{-(\alpha +n/2)}}$ ${\displaystyle {\mathrm{\Gamma }}_p}$ is themultivariate gamma function.
CDF	No analytic expression
Mean	${\displaystyle \mathbf{M}}$ if${\displaystyle \alpha >p/2}$, else undefined
Variance	${\displaystyle \frac{2(\mathbf{\Sigma }\otimes \mathbf{\Omega })}{\beta (2\alpha -p-1)}}$ if${\displaystyle \alpha >(p+1)/2}$, else undefined
CF	see below

An alternative parameterisation of the matrixt-distribution uses two parameters${\displaystyle \alpha }$ and${\displaystyle \beta }$ in place of${\displaystyle \nu }$.^[3]

This formulation reduces to the standard matrixt-distribution with${\displaystyle \beta =2,\alpha =\frac{\nu +p-1}{2}.}$

This formulation of the matrixt-distribution can be derived as thecompound distribution that results from an infinitemixture of a matrix normal distribution with aninverse multivariate gamma distribution placed over either of its covariance matrices.

If${\displaystyle \mathbf{X}\sim {\mathrm{T}}_{n,p}(\alpha ,\beta ,\mathbf{M},\mathbf{\Sigma },\mathbf{\Omega })}$ then^[2][3]

${\displaystyle {\mathbf{X}}^{\mathrm{T}}\sim {\mathrm{T}}_{p,n}(\alpha ,\beta ,{\mathbf{M}}^{\mathrm{T}},\mathbf{\Omega },\mathbf{\Sigma }).}$

The property above comes fromSylvester's determinant theorem:

${\displaystyle det\left({\mathbf{I}}_n+\frac{\beta }{2}{\mathbf{\Sigma }}^{-1}(\mathbf{X}-\mathbf{M}){\mathbf{\Omega }}^{-1}(\mathbf{X}-\mathbf{M}{)}^{\mathrm{T}}\right)=}$

${\displaystyle det\left({\mathbf{I}}_p+\frac{\beta }{2}{\mathbf{\Omega }}^{-1}({\mathbf{X}}^{\mathrm{T}}-{\mathbf{M}}^{\mathrm{T}}){\mathbf{\Sigma }}^{-1}({\mathbf{X}}^{\mathrm{T}}-{\mathbf{M}}^{\mathrm{T}}{)}^{\mathrm{T}}\right).}$

If${\displaystyle \mathbf{X}\sim {\mathrm{T}}_{n,p}(\alpha ,\beta ,\mathbf{M},\mathbf{\Sigma },\mathbf{\Omega })}$ and${\displaystyle \mathbf{A}(n\times n)}$ and${\displaystyle \mathbf{B}(p\times p)}$ arenonsingular matrices then^[2][3]

${\displaystyle \mathbf{A}\mathbf{X}\mathbf{B}\sim {\mathrm{T}}_{n,p}(\alpha ,\beta ,\mathbf{A}\mathbf{M}\mathbf{B},\mathbf{A}\mathbf{\Sigma }{\mathbf{A}}^{\mathrm{T}},{\mathbf{B}}^{\mathrm{T}}\mathbf{\Omega }\mathbf{B}).}$

Thecharacteristic function is^[3]

${\displaystyle {\varphi }_T(\mathbf{Z})=\frac{\exp (\mathrm{t}\mathrm{r}(i{\mathbf{Z}}^{\prime }\mathbf{M}))|\mathbf{\Omega }{|}^{\alpha }}{{\mathrm{\Gamma }}_p(\alpha )(2\beta {)}^{\alpha p}}|{\mathbf{Z}}^{\prime }\mathbf{\Sigma }\mathbf{Z}{|}^{\alpha }{B}_{\alpha }\left(\frac{1}{2\beta }{\mathbf{Z}}^{\prime }\mathbf{\Sigma }\mathbf{Z}\mathbf{\Omega }\right),}$

where

${\displaystyle B_{\delta }(\mathbf{W}\mathbf{Z})=|\mathbf{W}{|}^{-\delta }{\int }_{\mathbf{S}>0}\exp \left(\mathrm{t}\mathrm{r}(-\mathbf{S}\mathbf{W}-{\mathbf{S}}^{\mathbf{-}\mathbf{1}}\mathbf{Z})\right)|\mathbf{S}{|}^{-\delta -\frac{1}{2}(p+1)}d\mathbf{S},}$

and where${\displaystyle B_{\delta }}$ is the type-twoBessel function of Herz^{[clarification needed]} of a matrix argument.

• Multivariatet-distribution
• Matrix normal distribution

• A C++ library for random matrix generator

• Random matrices
• Multivariate continuous distributions

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