  We put forward and prove a simple theorem stating that the eigenvalues of a
tridiagonal matrix change their sign (as a set), once the signs of the diagonal
elements of the matrix are changed. We also provide an example of application
of this theorem in quantum physics. Specifically, we introduce the notion of
duality and self-duality for a sextic-polynomial quasi-exactly-solvable
potential, and demonstrate that the algebraic parts of the spectrum of the dual
potentials have signs opposite to one another (as sets). Our Theorem furnishes
an elegant one-line proof of this statement. In addition, we also prove it by
purely quantum-mechanical means - a far less straightforward method that
requires some effort.
