Given a matrix $[min(s,t]_{st}$ how can it be represented in terms of matrix operations?
For scalars, $min(s,t) = \frac{1}{2}(s + t - |s - t|)$.
The objective here is to find the conditional covariance and mean of a Brownian motion given $W_t = w_t,\ 0 \le t \le 1$
You might find the following useful: if $M$ is a matrix with entries $M_{s,t} = \min\{s,t\}$, then we have $M = AA^T$ with $A$ equal to the lower triangular matrix $$ A = \pmatrix{1\\1&1\\ \vdots & & \ddots \\1& \cdots &&1}. $$ It is also notable that $M^{-1}$ has the nice form $$ M^{-1} = \pmatrix{1&-2\\-2&\ddots & \ddots\\&\ddots&&-2\\&&-2&1}. $$