  In this paper we study sequences of vector orthogonal polynomials. The vector
orthogonality presented here provides a reinterpretation of what is known in
the literature as matrix orthogonality. These systems of orthogonal polynomials
satisfy three-term recurrence relations with matrix coefficients that do not
obey to any type of symmetry. In this sense the vectorial reinterpretation
allows us to study a non-symmetric case of the matrix orthogonality. We also
prove that our systems of polynomials are indeed orthonormal with respect to a
complex measure of orthogonality. Approximation problems of Hermite-Pad\'e type
are also discussed. Finally, a Markov's type theorem is presented.
