Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints

By J. William Helton, R. Far, Roland Speicher. 2007

120 citations recorded by the source corpus.

We show that the quadratic matrix equation $VW + \eta (W)W = I$, for given $V$ with positive real part and given analytic mapping $\eta$ with some positivity preserving properties, has exactly one solution $W$ with positive real part. We point out the relevance of this result in the context of operator-valued free probability theory and for the determination of the asymptotic eigenvalue distribution of band or block random matrices. We also address the problem of a numerical determination of the solution.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9b4bd036a01e88335640d64f2f07ea80394caff8

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In Applied Mathematics

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Citations in the corpus

Free Probability and Random Matrices (2014, background): “Note also that in [95] it was shown that there exists exactly one solution of the fixed point equation (9.”
Convergence of the largest singular value of a polynomial in independent Wigner matrices (2011, methodology): “We have been significantly influenced by the paper [13] which explored geometry and numerical analysis of the Schwinger–Dyson equation, and which could serve uninitiated readers as an introduction to the use of matricial semicircular elements.”
Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane (2015, background): “1) has been initiated so far that goes beyond establishing existence, uniqueness, and regularity in the regime where the parameter ́ is away from the real axis [1,23,26,35].”