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A P-matrix is a square matrix $X$ such that all principal submatrices of $X$have positive determinant.  Such matrices appear naturally in instances of the linear complementarityproblem, where these are precisely the matrices for which the correspondinglinear complementarity problem has a unique solution for any input vector.  Testing whether or not a square matrix is a P-matrix is co-NP complete, sowhile it is possible to exhibit polynomially-sized witnesses for the fact thata matrix is not a P-matrix, it is believed that there is no efficient way toprove that a given matrix is a P-matrix.  We will show that several well known witnesses for the fact that a matrix isnot a P-matrix are computationally equivalent, so that we are able to convertbetween them in polynomial time, answering a question raised inarXiv:1811.03841 .

On the computational equivalence of co-NP refutations of a matrix being a P-matrix