# Distribution for sum of Wishart Matrices with scale matrices proportional to identity

Consider the independent $n\times p$ dimensional matrices $\mathbf{X}_i, i=1,...,G$, from the matrix variate normal distribution, $N_{n,p}\left(0,\mathbf{I}_n, \mathbf{\Sigma}_i\right)$:

$$P_i\left(\mathbf{X}_i\right)\propto \exp\left[\operatorname{tr}\mathbf{X}_i\mathbf{\Sigma}_i^{-1}\mathbf{X}_i^{T}\right],$$ where $\mathbf{\Sigma}_i=\sigma^2\mathbf{I}_p$. Then $p\times p$ matrices $\mathbf{W}_i=\mathbf{X}_i^T\mathbf{X}_i$, are respectively from the wishart distribution $W_i\sim W_p\left(n,\Sigma_i\right)$. The matrix $\bar{\mathbf{W}}$ can be written as

$$\bar{\mathbf{W}}=\sum_{i=1}^G\frac{a_i}{n}\mathbf{X}_i^T\mathbf{X}_i= \sum_{i=1}^G\mathbf{X}_i^T\left(\frac{a_i}{n}\mathbf{I}_n\right)\mathbf{X}_i= \left[\begin{matrix} \mathbf{X}_1^T & \mathbf{X}_2^T & \ldots & \mathbf{X}_G^T \end{matrix}\right] \left[\begin{matrix} \frac{a_1}{n}\mathbf{I}_n & 0 & \ldots & 0\\ 0 & \frac{a_2}{n}\mathbf{I}_n & \ldots& 0\\ \vdots&&\ddots&\vdots\\ 0&0&\ldots&\frac{a_G}{n}\mathbf{I}_n \end{matrix}\right] \left[\begin{matrix} \mathbf{X}_1 \\ \mathbf{X}_2\\ \vdots\\ \mathbf{X}_G \end{matrix}\right]=\mathbf{X}^T\mathbf{D}\mathbf{X}=\mathbf{Y}^T\mathbf{Y}$$

Where we define

$$\mathbf{X}^T=\left[\begin{matrix} \mathbf{X}_1^T & \mathbf{X}_2^T & \ldots & \mathbf{X}_G^T \end{matrix}\right],\\ \mathbf{D}=\operatorname{diag} \left(\frac{a_1}{n}\mathbf{I}_n,\ldots,\frac{a_G}{n}\mathbf{I}_n\right),\\ \mathbf{Y}=\mathbf{D}^{1/2}\mathbf{X}.$$ With the above information, I wanna show that $\mathbf{Y}$ satisfy the distribuion $$P_Y\left(\mathbf{Y}\right)\propto \exp\left[\operatorname{tr}\mathbf{Y}^T\mathbf{V}^{-1}\mathbf{Y}\right],$$ where $\mathbf{V}=\operatorname{diag} \left(\frac{a_1}{n}\sigma^2\mathbf{I}_n,\ldots,\frac{a_G}{n}\sigma^2\mathbf{I}_n\right)$. Therefore, we are looking essentially at a semicorrelated Wishart case, with the diagonal-covariance matrix possessing some multiplicities.

Basically, I want to find the distribution for $\bar{\mathbf{W}}$.

This follows Kumar et al. (2015): On the Exact and Approximate Eigenvalue Distribution for Sum of Wishart Matrices. All credit to them.