# Marginal Distribution of Partition Matrix

Suppose that $$X\sim IW_{p}(n,I_p)$$ has an inverse Wishart distribution, which probability density function is $$f(X\mid n)\propto |X|^{-\frac{n+p+1}{2}}exp\Big(-\frac{1}{2}tr( X^{-1})\Big),~~\qquad (1)$$ Partition the matrices $$X$$ with $$X=\pmatrix{X_{11}&X_{12}\\X_{21}&X_{22}},$$ where $$X_{ij}$$ is $$p_i\times p_j$$ matrices. From the definition of IW, we know $$X_{11}\sim IW_{p_1}(n-p_2,I_{p_1}).$$

Qustion: If random $$p\times p$$ positive matrice $$Y$$ satisfies $$f(Y\mid n)\propto |Y|^{-\frac{n+p+1}{2}}exp\Big(-\frac{1}{2}tr( Y^{-1})\Big)\Big[\prod_{i=1}^p(\lambda_i-\lambda_j)\Big]^{-1},~~\qquad (2)$$ where $$\lambda_1>\lambda_2>\cdots>\lambda_p>0$$ are the ordered eigenvalues.

Partition the matrices $$Y$$ with $$Y=\pmatrix{Y_{11}&Y_{12}\\Y_{21}&Y_{22}}.$$ What is the marginal distribution of $$Y_{11}???$$