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On the asymptotic behaviour of the spectra of non-symmetric random (0,1) matrices

✍ Scribed by Ferenc Juhász

Publisher
Elsevier Science
Year
1982
Tongue
English
Weight
373 KB
Volume
41
Category
Article
ISSN
0012-365X

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✦ Synopsis

In [2] we investigated the spectrum of a random gl,tph (symmetric matrix). In the present paper we are going to show that the results of [2] carry over to non-symmetric random (Q, 1) matrices (directed graphs), namely the largest eigenvalue is of order n, while the other eigenvalues xre of order n 1'2fF (E > 0 arbitrary). The method used also applies in the symmetric case.

Let A, = (aij) be an n X n matrix whose entries me, for i # j, independent identically distributed random variables, P(&j = l)==p* P(u+=O)=q=l-p.

Assume further that aii E 0.

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Asymptotic distribution of the even and
Asymptotic distribution of the even and odd spectra of real symmetric Toeplitz matrices
✍ William F. Trench 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 87 KB

If n t rÀs n rYs0 is a real symmetric Toeplitz (RST) matrix then R n has a basis consisting of dna2e eigenvectors x satisfying (A) tx x and na2 eigenvectors y satisfying (B) ty Ày, where t is the ¯ip matrix. We say that an eigenvalue k of n is even if a k-eigenvector of n satis®es (A), or odd if a k