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This paper is concerned with the reduction of a unitary matrix U to CMV-likeshape. A Lanczos--type algorithm is presented which carries out the reductionby computing the block tridiagonal form of the Hermitian part of U, i.e., ofthe matrix U+U^H. By elaborating on the Lanczos approach we also propose analternative algorithm using elementary matrices which is numerically stable. IfU is rank--structured then the same property holds for its Hermitian part and,therefore, the block tridiagonalization process can be performed using therank--structured matrix technology with reduced complexity. Our interest in theCMV-like reduction is motivated by the unitary and almost unitary eigenvalueproblem. In this respect, finally, we discuss the application of the CMV-likereduction for the design of fast companion eigensolvers based on the customaryQR iteration.

Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding