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We construct rational all-pass matrix functions with real-valued coefficientsfor mirroring pairs of complex-conjugated determinantal roots of a rationalmatrix. This problem appears, for example, when proving the spectralfactorization theorem, or, more recently, in the literature on possiblynon-invertible or possibly non-causal vector autoregressive moving average(VARMA) models. In general, it is not obvious whether the all-pass matrixfunction (and as a consequence the all-pass transformed rational matrix withinitally real-valued coefficients) which mirrors complex-conjugated roots atthe unit circle has real-valued coefficients. Naive constructions result inall-pass functions with complex-valued coefficients which implies that thereal-valued parameter space (usually relevant for estimation) is left.

All-Pass Functions for Mirroring Pairs of Complex-Conjugated Roots of Rational Matrix Functions