On the first degree Fejér-Riesz factorization and its applications to X + A^⁎X⁻¹A = Q

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• 作者：Moody T. Chu¹ ; ^{chu@math.ncsu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：39B42 ; 15A23 ; 15A24 ; 47J25 ; 49M15 ; 65J15
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：489
• 期：Complete
• 页码：123-143
• 全文大小：520 K

文摘

Given a Laurent polynomial with matrix coefficients that is positive semi-definite over the unit circle in the complex plane, the Fejér–Riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint. This paper exploits such a factorization in its simplest form of degree one and its relationship with the nonlinear matrix equation X+A^⁎X⁻¹A=Q$X+A^{⁎}{X}^{-1}A=Q$. In particular, the nonlinear equation can be recast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation is readily obtainable from hermitian eigenvalue computation. The unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate-free Newton iteration for high precision calculation. This approach offers a complete parametrization of all solutions and, in contrast to most existent algorithms, makes it possible to find all solutions if so desired.