Real skew-symmetric and periodic skew-symmetric tridiagonal matrices with prescribed spectral data

Skew-symmetric matrices have various applications in physics, one of which is in the gyroscopic system. In this paper, we retrieve a unique real skew-symmetric tridiagonal matrix from the maximal and minimal imaginary parts of the eigenvalues of all its leading principal submatrices. For n even, we then reconstruct an n-by-n real periodic skew-symmetric tridiagonal matrix from the imaginary parts of all its eigenvalues, and those of the leading principal submatrix of order $n-1$, and a given positive number. The necessary and sufficient conditions for the existence of such matrices are given, and we show that the total number of possible n-by-n periodic skew-symmetric tridiagonal matrices is at most $2^{\lfloor \frac{n-s}{2}\rfloor }$, where s is the number of common elements in the two prescribed spectra. The proofs of the obtained results provide algorithmic procedures for the aimed reconstructions, which are supported by some illustrative numerical experiments.