# Some Matrix-Variate Models Applicable in Different Areas

## Abstract

**:**

## 1. Introduction

## 2. Evaluation of Some Matrix-Variate Integrals and the Resulting Models

#### 2.1. Notations and Formats

#### 2.2. Some Matrix-Variate Integrals

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

#### Evaluation of the Integral in (5) in the Real Case and (6) in the Complex Case

#### 2.3. Evaluation of the Sine and Cosine Product in the Real Case

#### Evaluation of the Integral over the ${\theta}_{j}$s in the Complex Case

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Remark**

**1.**

**Theorem**

**4.**

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Theorem**

**5.**

**Theorem**

**6.**

## 3. Some Integrals Involving Type 2 Beta Forms

**Theorem**

**7.**

**Theorem**

**8.**

**Theorem**

**9.**

**Theorem**

**10.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Theorem**

**13.**

**Theorem**

**14.**

**Remark**

**2.**

**Theorem**

**15.**

**Proof.**

**Remark**

**3.**

**Theorem**

**16.**

## 4. Matrix-Variate Type 1 Beta Forms

**Theorem**

**17.**

**Theorem**

**18.**

**Theorem**

**19.**

**Theorem**

**20.**

**Remark**

**4.**

**Theorem**

**21.**

**Proof.**

**Theorem**

**22.**

**Theorem**

**23.**

**Theorem**

**24.**

**Theorem**

**25.**

**Theorem**

**26.**

## 5. Concluding Remarks

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof.**

**Step 1.**Consider a vector case first. Let X be a $p\times 1$ real vector, ${X}^{\prime}=[{x}_{1},\dots ,{x}_{p}]$, where a prime denotes the transpose, and ${x}_{j}^{\prime}s$ are distinct real scalar variables. Consider the following:$$Y=AX\Rightarrow \left[\begin{array}{c}{y}_{1}\\ {y}_{2}\\ \vdots \\ {y}_{p}\end{array}\right]=\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1p}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ {a}_{p1}& {a}_{p2}& \dots & {a}_{pp}\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ \vdots \\ {x}_{p}\end{array}\right]$$

**Step 2.**Now, consider the $p\times q$ matrices Y and X, and let A be a nonsingular $p\times p$ constant matrix. Then, we have the following:$$Y=AX\Rightarrow [{Y}_{(1)},\dots ,{Y}_{(q)}]=[A{X}_{(1)},\dots ,A{X}_{(q)}]\Rightarrow {Y}_{(i)}=A{X}_{(i)}$$$$U=\left[\begin{array}{c}{Y}_{(1)}\\ {Y}_{(2)}\\ \vdots \\ {Y}_{(q)}\end{array}\right],\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}V=\left[\begin{array}{c}{X}_{(1)}\\ {X}_{(2)}\\ \vdots \\ {X}_{(q)}\end{array}\right]\to (\frac{\partial U}{\partial V})=\left[\begin{array}{cccc}A& O& \dots & O\\ O& A& \dots & O\\ \vdots & \vdots & \ddots & \vdots \\ O& O& \dots & A\end{array}\right]\Rightarrow \mathrm{d}Y={|A|}^{q}\mathrm{d}X$$**Step 3.**Now, consider $Y=XB$, take the rows of Y and X, and proceed as in Case 2 above to see that $\mathrm{d}Y={|B|}^{p}\mathrm{d}X$. Now, consider $Y=AXB\Rightarrow Y=AZ,Z=XB\Rightarrow \mathrm{d}Y={|A|}^{q}\mathrm{d}Z,\mathrm{d}Z={|B|}^{p}\mathrm{d}X$. Now, the lemma is proven. □

## Appendix B

- The Theorem 2, part (2) normalizing constant from [6] is as follows:$$\begin{array}{cc}\hfill {C}_{2}& =\frac{{C}_{0}s{\Gamma}_{m}(\frac{n}{2}){r}^{\frac{m}{s}(q-\frac{m+1}{2}+\frac{n}{2})}}{{\pi}^{\frac{mn}{2}}{\Gamma}_{m}(\frac{1}{s}(q-\frac{m+1}{2}+\frac{n}{2}))}=\frac{{C}_{0}\delta {\Gamma}_{p}(\frac{q}{2}){\alpha}^{\frac{p}{\delta}(\gamma +\frac{q}{2})}}{{\pi}^{\frac{pq}{2}}{\Gamma}_{p}(\frac{1}{\delta}(\gamma +\frac{q}{2}))}\hfill \\ \hfill \text{Correct one}& =\frac{{C}_{0}\delta {\Gamma}_{p}(\frac{q}{2}){\alpha}^{\frac{p}{\delta}(\gamma +\frac{q}{2})}\Gamma (p(\gamma +\frac{q}{2}))}{{\pi}^{\frac{pq}{2}}{\Gamma}_{p}(\gamma +\frac{q}{2})\Gamma (\frac{p}{\delta}(\gamma +\frac{q}{2}))}.\hfill \end{array}$$
- The Theorem 3, part (1) normalizing constant ${C}_{3}$ from [6], translated in terms of the parameters of our Theorem 3 and the correct normalizing constant, results in the following:$$\begin{array}{cc}\hfill {C}_{3}& =\frac{\rho \Gamma (\frac{m\alpha}{2}){r}^{\frac{1}{\rho}(\beta -1+\frac{m\alpha}{2})}}{{\Gamma}_{m}(\frac{\alpha}{2})\Gamma (\frac{1}{\rho}(\beta -1+\frac{m\alpha}{2}))}=\frac{\delta \Gamma (\frac{pq}{2}){\alpha}^{\frac{1}{\delta}(\eta +\frac{pq}{2})}}{{\Gamma}_{p}(\frac{q}{2})\Gamma (\frac{1}{\delta}(\eta +\frac{pq}{2}))}\hfill \\ \hfill \text{Correct one}& =\frac{\delta {\Gamma}_{p}(\frac{q}{2}){\alpha}^{\frac{1}{\delta}(p(\gamma +\frac{q}{2})+\eta )}\Gamma (p(\gamma +\frac{q}{2}))}{{\pi}^{\frac{pq}{2}}{\Gamma}_{p}(\gamma +\frac{q}{2})\Gamma (\frac{1}{\delta}(p(\gamma +\frac{q}{2})+\eta ))}.\hfill \end{array}$$

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**MDPI and ACS Style**

Mathai, A.M.
Some Matrix-Variate Models Applicable in Different Areas. *Axioms* **2023**, *12*, 936.
https://doi.org/10.3390/axioms12100936

**AMA Style**

Mathai AM.
Some Matrix-Variate Models Applicable in Different Areas. *Axioms*. 2023; 12(10):936.
https://doi.org/10.3390/axioms12100936

**Chicago/Turabian Style**

Mathai, Arak M.
2023. "Some Matrix-Variate Models Applicable in Different Areas" *Axioms* 12, no. 10: 936.
https://doi.org/10.3390/axioms12100936