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Almost skew-symmetric matrices are real matrices whose symmetric parts have rank one. Using the notion of the numerical range, we obtain eigenvalue inequalities and a localization of the spectrum of an almost skew-symmetric matrix. We show that almost skew-symmetry is invariant under principal pivot transformation and inversion, and that the symmetric parts of Schur complements in almost skewsymmetric matrices have rank at most one. We also use affine combinations of A and A t to gain furtherdoi:10.1216/rmjm/1181069905 fatcat:2wjcomx7pvb6jawdyzvye7so4u