Probabilistic factorization of a quadratic matrix polynomial

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Abstract
A purely algebraic result. We begin by stating the following theorem. Theorem. Let E be a finite set, and letdenote the set of real E × E matrices with non-negative off-diagonal elements and with non-positive row sums. Let A be a symmetric element of, and let V be a diagonal real E × E matrix. Then there exists a unique pair (H+, H−) of elements ofsuch thatI denoting the identity E × E matrix, and the superscript T signifying transpose. It is an immediate consequence that

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