  An $n\times n$ matrix is said to have a self-interlacing spectrum if its
eigenvalues $\lambda_k$, $k=1,\ldots,n$, 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.
