Self-interlacing polynomials II: Matrices with self-interlacing spectrum

An $n\times n$ matrix is said to have a self-interlacing spectrum if its eigenvalues ${\lambda }_k$, $k=1,\dots ,n$, are distributed as follows $${\lambda }_1>-{\lambda }_2>{\lambda }_3>\cdots >(-1{)}^{n-1}{\lambda }_n>0.$$ A method for constructing sign definite matrices with self-interlacing spectra from totally nonnegative ones is presented. We apply this method to bidiagonal and tridiagonal matrices. In particular, we generalize a result by O. Holtz on the spectrum of real symmetric anti-bidiagonal matrices with positive nonzero entries.