Measurable diagonalization of positive definite matrices

In this paper we show that any positive definite matrix V with measurable entries can be written as V=U@LU^*, where the matrix @L is diagonal, the matrix U is unitary, and the entries of U and @L are measurable functions (U^* denotes the transpose conjugate of U). 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.

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