Spectra of Tridiagonal Matrices over a Field

Abstract

We consider spectra of $n$-by-$n$ irreducible tridiagonal matrices over a field and of their $n-1$-by-$n-1$ trailing principal submatrices. The real symmetric and complex Hermitian cases have been fully understood: it is necessary and sufficient that the necessarily real eigenvalues are distinct and those of the principal submatrix strictly interlace. So this case is very restrictive. By contrast, for a general field, the requirements on the two spectra are much less restrictive. In particular, in the real or complex case, the $n$-by-$n$ characteristic polynomial is arbitrary (so that the algebraic multiplicities may be anything in place of all 1's in the classical cases) and that of the principal submatrix is the complement of a lower dimensional algebraic set (and so relatively free). Explicit conditions are given.