I'm constructing an affinity/similarity matrix $\mathbf{A}_{n,n}$ which must express some sort of relationship among a set size $n$ of random vectors $\mathbf{x_i}$ with unknown distribution $F$. For any two feature vectors $\mathbf{x_i, x_j}$ and matrix $\mathbf{B}$, which is learned/trained using the data, I compute similarity between any two $\mathbf{x_i, x_j}$:
$a_{i,j} = \mathbf{(B^{T}x_i)^{T}B^{T}x_j=x_i^{T}BB^{T}x_{j}} $
Can I prove at least some property of $\mathbf{A}$, e.g. all $a_{ij} \geq 0$, given that none of the inputs is guaranteed to be positive. Or, since $\mathbf{x_i}$ are random, something about the distribution of $a_{ij}$?
Links to publications and/or relevant theorems will do. This is for my research on Deep Learning with CT scans used for COVID-19 prediction.