Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-020-00947-2 Matrix Uvarov Transformation on the Unit Circle: Asymptotic Properties Herbert Dueñas 1 · Edinson Fuentes 1 · Luis E. Garza 2 Received: 17 May 2019 / Revised: 7 May 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020 Abstract Let σ be an l × l Hermitian matrix measure supported on the unit circle. In this contribution, we study some algebraic and analytic properties of matrix orthogonal polynomials associated with the Uvarov matrix transformation of σ defined by dσ u m (z ) = dσ(z ) + m  j =1 M j δ(z − ζ j ), where M j is an l × l positive definite matrix, ζ j ∈ C with ζ j  = ζ i and δ is the Dirac matrix measure. Keywords Matrix orthogonal polynomials on the unit circle · Uvarov matrix transformation · Relative asymptotics · Zeros Mathematics Subject Classification 33C45 · 33D45 · 42C05 1 Introduction Let D ={z ∈ C :|z | < 1} be the open disc, and let μ be a non-trivial measure sup- ported on T =  z ∈ C : z = e i θ ,θ ∈ (−π,π ]  . It is a very well-known fact (see Communicated by Ali Hassan Mohamed Murid. B Luis E. Garza luis_garza1@ucol.mx Herbert Dueñas haduenasr@unal.edu.co Edinson Fuentes efuentes@unal.edu.co 1 Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia 2 Facultad de Ciencias, Universidad de Colima, Colima, Mexico 123