= Jump matrix
{title2=$J=M_-^{-1}M_+$}
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> satisfies
$$
M_+(z)=M_-(z)J(z).
$$
Matrix 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>.<Riemann-Hilbert Problems by Percy Deift>[Lecture 1, printed pages 5–6, oriented sides and jump equation]{ref}
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