Measurable diagonalization of positive definite matrices

In this paper we show that any positive definite matrix VV with measurable entries can be written as V=UΛU∗V=UΛU∗, where the matrix ΛΛ is diagonal, the matrix UU is unitary, and the entries of UU and ΛΛ are measurable functions (U∗U∗ denotes the transpose conjugate of UU).This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials.