-spectral factorization for rational matrix functions with alternative realization.
In this study various Jacobians of transformations of singular random matrices are found. An alternative proof of Uhlig's first conjecture (Uhlig (1994)) is proposed. Furthermore, we propose various extensions of this conjecture under different singularities. Finally, an application of the theory of singular distributions is discussed.
We give a sufficient condition under which any Jordan automorphism of a triangular algebra is either an automorphism or an anti-automorphism.