斜对称模式随机矩阵的体行为
Bulk behaviour of skew-symmetric patterned random matrices
摘要：
实对称模式矩阵的极限谱分布（LSD）已经得到了很好的研究。本文研究了斜对称/反对称模式随机矩阵，建立了几种常见矩阵的最小二乘估计。对于反对称Wigner、反对称Toeplitz和反对称循环，最小二乘方差（在虚轴上）与对称情形相同。然而，对于反对称Hankel和反对称逆循环，我们得到了新的LSD。我们还证明了这些矩阵的三角形版本的LSD的存在性。然后，我们通过改变矩阵下三角部分的符号来引入对对称矩阵的相关修改。在这种情况下，修改的Wigner，修改的Hankel和修改的反向循环具有相同的LSD作为其通常的对称对应物，而新的LSD获得修改的Toeplitz和修改的对称循环。
英文摘要：
Limiting Spectral Distributions (LSD) of real symmetric patterned matrices have been well-studied. In this article, we consider skew-symmetric/anti-symmetric patterned random matrices and establish the LSDs of several common matrices. For the skew-symmetric Wigner, skew-symmetric Toeplitz and the skew-symmetric Circulant, the LSDs (on the imaginary axis) are the same as those in the symmetric cases. For the skew-symmetric Hankel and the skew-symmetric Reverse Circulant however, we obtain new LSDs. We also show the existence of the LSDs for the triangular versions of these matrices. We then introduce a related modification of the symmetric matrices by changing the sign of the lower triangle part of the matrices. In this case, the modified Wigner, modified Hankel and the modified Reverse Circulants have the same LSDs as their usual symmetric counterparts while new LSDs are obtained for the modified Toeplitz and the modified Symmetric Circulant.
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