Suppose $X_1, \cdots, X_I$ are continuous random variables from unspecified distributions $F_1, \cdots, F_I$. Let $m_{ij} = \mathbb{P}(X_i < X_j)$ and $M = [m_{ij}]_{ij}$ (that is, $M$ is a matrix population with entries $m_{ij}$. What are some properties of this matrix? Specifically, I'm most interested in eigenvalues, determinants, etc. Some immediate properties I can derive are
- $M_{ii} = \frac{1}{2}$
- $M + M^\intercal = \mathbf{1}\mathbf{1}^\intercal$
- $\text{tr}(M) = \frac{I}{2}$
- $\mathbf{v}^\intercal M \mathbf{v} = \frac{1}{2} \mathbf{v}^\intercal\mathbf{1}\mathbf{1}^\intercal\mathbf{v}$
What are some others?