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The purpose of this note is to establish, in terms of the primarycoefficients in the framework of the tridiagonal theory developed by Delsarteand Genin in the environment of nonnegative definite Toeplitz matrices,necessary and sufficient conditions for the monotonicity with respect to a realparameter of zeros of paraorthogonal polynomials on the unit circle. It is alsoprovided tractable sufficient conditions and an application example. Thesepolynomials can be regarded as the characteristic polynomials of any matrixsimilar to an unitary upper Hessenberg matrix with positive subdiagonalelements.

On monotonicity of zeros of paraorthogonal polynomials on the unit circle