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

摘要

An $nimes n$ 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 spectrum from totally nonnegative ones is presented. This method is applied to bidiagonal and tridiagonal matrices. In particular, a result by O. Holtz on the spectrum of real symmetric anti-bidiagonal matrices with positive nonzero entries is generalized.