We provide necessary and sufficient conditions for the matrix equation X J AX " B to be consistent when B is a symmetric matrix, for all matrices A with a few exceptions. The matrices A, B, and X (unknown) are matrices with complex entries. We first see that we can restrict ourselves to the case where A and B are given in canonical form for congruence and, then, we address the equation with A and B in such form. The characterization strongly depends on the canonical form for congruence of A. The problem we solve is equivalent to: given a complex bilinear form (represented by A) find the maximum dimension of a subspace such that the restriction of the bilinear form to this subspace is a symmetric non-degenerate bilinear form.

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A list of complex numbers is realizable if it is the spectrum of a nonnegative matrix. In 1949 Suleimanova posed the nonnegative inverse eigenvalue problem (NIEP): the problem of determining which lists of complex numbers are realizable. The version for reals of the NIEP (RNIEP) asks for realizable lists of real numbers. In the literature there are many sufficient conditions or criteria for lists of real numbers to be realizable. We will present an unified account of these criteria. Then we will see that the decision problem associated to the RNIEP is NP-hard and we will study the complexity for the decision problems associated to known criteria.Comment: 10 pages, 1 figur

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