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<p>Intertwining analysis, optimization, numerical analysis and algebra, computing conjugate co-gradients of real-valued quotients gives rise to eigenvalue problems. In the linear Hermitian case, by inspecting optimal quotients in terms of taking the conjugate co-gradient for their critical points, a generalized folded spectrum eigenvalue problem arises. Replacing the Euclidean norm in optimal quotients with the <i>p</i>-norm, a matrix version of the so-called <i>p</i>-Laplacian eigenvalue problem arises. Such nonlinear eigenvalue problems seem to be naturally classified as being a special case of homogeneous problems. Being a quite general class, tools are developed for recovering whether a given homogeneous eigenvalue problem is a gradient eigenvalue problem. It turns out to be a delicate issue to come up with a valid quotient. A notion of nonlinear Hermitian eigenvalue problem is suggested. Cauchy–Schwarz quotients are introduced to a have a way to approach non-gradient eigenvalue problems.</p>

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Gradients of quotients and eigenvalue problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月23日

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  • Marko Huhtanen,
  • Olavi Nevanlinna

摘要

Intertwining analysis, optimization, numerical analysis and algebra, computing conjugate co-gradients of real-valued quotients gives rise to eigenvalue problems. In the linear Hermitian case, by inspecting optimal quotients in terms of taking the conjugate co-gradient for their critical points, a generalized folded spectrum eigenvalue problem arises. Replacing the Euclidean norm in optimal quotients with the p-norm, a matrix version of the so-called p-Laplacian eigenvalue problem arises. Such nonlinear eigenvalue problems seem to be naturally classified as being a special case of homogeneous problems. Being a quite general class, tools are developed for recovering whether a given homogeneous eigenvalue problem is a gradient eigenvalue problem. It turns out to be a delicate issue to come up with a valid quotient. A notion of nonlinear Hermitian eigenvalue problem is suggested. Cauchy–Schwarz quotients are introduced to a have a way to approach non-gradient eigenvalue problems.