 Jump matrix

A jump matrix is a prescribed invertible matrix-valued function $J$ on an oriented contour in a matrix Riemann-Hilbert problem. With plus boundary on the left of the contour, the unknown holomorphic matrix function satisfiesMatrix multiplication order matters. Where $M_{-}$ is invertible, $J=M_{-}^{-1}{M}_{+}$, rather than $M_{+}{M}_{-}^{-1}$ in general. Reversing contour orientation swaps the sides and replaces $J$ by $J^{-1}$.

For the positively oriented unit circle, a constant invertible matrix $J=A$ is realized by $M=A$ inside and $M=I$ outside. The exterior solution is normalized to $I$ at infinity. More complicated jumps encode boundary values of spectral or scattering data; prescribed poles, endpoint behavior and normalization must also be specified to define the full Riemann-Hilbert problem.^[ref]