  Pseudo-unitary circuits are recurring in both S-matrix theory and analysis of
No-Go theorems. We propose a matrix and diagrammatic representation for the
operation that maps S-matrices to T-matrices and, consequently, a unitary group
to a pseudo-unitary one. We call this operation ``partial inversion'' and show
its diagrammatic representation in terms of permutations. We find the
expressions for the deformed metrics and deformed dot products that preserve
physical constraints after partial inversion. Subsequently, we define a special
set that allows for the simplification of expressions containing infinities in
matrix inversion. Finally, we propose a renormalized-growth algorithm for the
T-matrix as a possible application. The outcomes of our study expand the
methodological toolbox needed to build a family of pseudo-unitary and
inter-pseudo-unitary circuits with full diagrammatic representation in three
dimensions, so that they can be used to exploit pseudo-unitary flexibilization
of unitary No-Go Theorems and renormalized circuits of large scattering
lattices.
