For $0 \le A \in GL(n,\mathbb{R})$, let $Aw = \Delta(A)$, where $\Delta$ denotes the map taking a matrix to a vector of its diagonal entries and/or forming a diagonal matrix from a vector, according to context. Writing $Z := \Delta(\Delta(A))^{-1}A$, we have the equation $Zw = 1$. Readers familiar with the theory of magnitude à la Leinster will recognize this as the equation defining a "weighting" from a "similarity matrix" $Z$ (hence the category theory tags). It turns out that if $\Delta(\Delta(A))^{-1} 11^T \ge A$ (i.e., the diagonal entries of $A$ are greater than or equal to the other entries in their row) and $A_{j\ell}A_{\ell k} \le A_{jj}A_{jk}$ identically (this last being a rephrasing of the triangle inequality), then writing $Z = \exp(-td)$ componentwise yields a Lawvere metric $d$ for each $t$, though the parametric dependence on $t$ is linear and hence boring.
(FWIW, the RHS of the equation $Aw = \Delta(A)$ is used to ensure that $\Delta(d) = 0$, which seems pretty important to any sensible sort of metric geometry, "metametrics" notwithstanding; the constraint $\Delta(\Delta(A))^{-1} 11^T \ge A$ is used to ensure that $d \ge 0$, which also seems pretty important.)
My question is:
are such matrices $A$ (better still, equations of the form above involving them) discussed in the literature? What about if we drop the "triangle inequality" part?
