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Eigenvalue contour lines of Kac–Murdock–Szegő matrices with a complex parameter
Linear Algebra and its Applications ( IF 1.1 ) Pub Date : 2021-07-26 , DOI: 10.1016/j.laa.2021.07.016
George Fikioris 1 , Christos Papapanos 1, 2
Affiliation  

A previous paper studied the so-called borderline curves of the Kac–Murdock–Szegő matrix Kn(ρ)=[ρ|jk|]j,k=1n, where ρC. These are the level curves (contour lines) in the complex-ρ plane on which Kn(ρ) has a type-1 or type-2 eigenvalue of modulus n, where n is the matrix dimension. Those curves have cusps at all critical points ρ=ρc at which multiple (double) eigenvalues occur. The present paper determines corresponding curves pertaining to eigenvalues of modulus νn. We find that these curves no longer present cusps; and that, when ν<n, the cusps have in a sense transformed into loops. We discuss the meaning of the winding numbers of our curves. Finally, we point out possible extensions to more general matrices.



中文翻译:

具有复参数的 Kac-Murdock-Szegő 矩阵的特征值等高线

之前的一篇论文研究了 Kac-Murdock-Szegő 矩阵的所谓边界曲线 n(ρ)=[ρ|j-|]j,=1n, 在哪里 ρC. 这些是复ρ平面中的水平曲线(等高线),在该平面上n(ρ)具有模数n的类型 1 或类型 2 特征值,其中n是矩阵维度。这些曲线在所有关键点都有尖点ρ=ρC出现多个(双)特征值。本文确定了与模量特征值有关的相应曲线νn. 我们发现这些曲线不再出现尖峰;那,当ν<n,从某种意义上说,尖端已经变成了循环。我们讨论曲线绕数的含义。最后,我们指出了对更一般矩阵的可能扩展。

更新日期:2021-07-30
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