Bulk behavior of Schur–Hadamard products of symmetric random matrices

2014 ◽  
Vol 03 (02) ◽  
pp. 1450007 ◽  
Author(s):  
Arup Bose ◽  
Soumendu Sundar Mukherjee
Keyword(s):  
Random Matrices ◽  
Hadamard Product ◽  
Hankel Matrices ◽  
Hadamard Products ◽  
Spectral Distributions ◽  
Patterned Matrices ◽  
Invariance Theorem ◽  
Bulk Behavior ◽  
General Method

We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur–Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSD of Schur–Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur–Hadamard product of independent Toeplitz and Hankel matrices has the semi-circular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples.

Some patterned matrices with independent entries

2020 ◽  
pp. 2150030
Author(s):  
Arup Bose ◽  
Koushik Saha ◽  
Priyanka Sen
Keyword(s):  
Random Matrices ◽  
Spectral Distribution ◽  
Finite Variance ◽  
Weighted Sum ◽  
Hankel Matrices ◽  
Special Cases ◽  
Common Thread ◽  
Different Types ◽  
Patterned Matrices ◽  
Variance Profile

Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the [Formula: see text]th moment of the limit equals a weighted sum over different types of pair-partitions of the set [Formula: see text] and are universal. Some results are also known for the sparse case. In this paper, we generalize these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.

Hadamard Products and Powers of TN Matrices

2011 ◽  
Author(s):  
Shaun M. Fallat ◽  
Charles R. Johnson
Keyword(s):  
Hadamard Product ◽  
Hadamard Products ◽  
Interesting Topic

This chapter introduces the Hadamard (or entrywise) product and offers a survey of the known results pertaining to Hadamard products of TN matrices, including a note on Hadamard powers. The well-known fact that TP matrices are not closed under Hadamard multiplication has certainly not deterred researchers from this interesting topic. In fact, it can be argued that this has fueled considerable investigations on when such closure issues, with particular attention being paid to specific subclasses of TP/TN matrices. Hence, the chapter first defines and denotes the Hadamard product of two m-by-n matrices A = [asubscript ij] and B = [bsubscript ij] before turning to a number of concepts connected with Hadamard products and Hadamard powers of TPₖ matrices.

Patterned sparse random matrices: A moment approach

2017 ◽  
Vol 06 (03) ◽  
pp. 1750011
Author(s):  
Debapratim Banerjee ◽  
Arup Bose
Keyword(s):  
Random Matrices ◽  
Limit Distribution ◽  
Spectral Distribution ◽  
Explicit Description ◽  
Circulant Matrices ◽  
Hankel Matrices ◽  
Largest Eigenvalue ◽  
Empirical Spectral Distribution ◽  
Moment Approach ◽  
The Largest Eigenvalue

We consider four specific [Formula: see text] sparse patterned random matrices, namely the Symmetric Circulant, Reverse Circulant, Toeplitz and the Hankel matrices. The entries are assumed to be Bernoulli with success probability [Formula: see text] such that [Formula: see text] with [Formula: see text]. We use the moment approach to show that the expected empirical spectral distribution (EESD) converges weakly for all these sparse matrices. Unlike the Sparse Wigner matrices, here the random empirical spectral distribution (ESD) converges weakly to a random distribution. This weak convergence is only in the distribution sense. We give explicit description of the random limits of the ESD for Reverse Circulant and Circulant matrices. As in the non-sparse case, explicit description of the limits appears to be difficult to obtain in the Toeplitz and Hankel cases. We provide some properties of these limits. We then study the behavior of the largest eigenvalue of these matrices. We prove that for the Reverse Circulant and Symmetric Circulant matrices the limit distribution of the largest eigenvalue is a multiple of the Poisson. For Toeplitz and Hankel matrices we show that the non-degenerate limit distribution exists, but again it does not seem to be easy to obtain any explicit description.

scholarly journals Identifiability of parametric random matrix models

2019 ◽  
Vol 22 (03) ◽  
pp. 1950018
Author(s):  
Tomohiro Hayase
Keyword(s):  
Probability Theory ◽  
Matrix Models ◽  
Random Matrices ◽  
Random Matrix ◽  
Free Probability ◽  
Parameter Identifiability ◽  
Free Probability Theory ◽  
Noise Type ◽  
Random Matrix Models ◽  
Spectral Distributions

We investigate parameter identifiability of spectral distributions of random matrices. In particular, we treat compound Wishart type and signal-plus-noise type. We show that each model is identifiable up to some kind of rotation of parameter space. Our method is based on free probability theory.

scholarly journals Limiting spectral distributions of sums of products of non-Hermitian random matrices

2018 ◽  
Vol 38 (2) ◽  
pp. 359-384
Author(s):  
Holger Kösters ◽  
Alexander Tikhomirov
Keyword(s):  
Probability Theory ◽  
Random Matrices ◽  
General Framework ◽  
Free Probability ◽  
Singular Value ◽  
Eigenvalue Distribution ◽  
Limiting Distributions ◽  
Free Probability Theory ◽  
Spectral Distributions ◽  
Sums Of Products

For fixed l≥0 and m≥1, let Xn0, Xn1,..., Xnl be independent random n × n matrices with independent entries, let Fn0 := Xn0, Xn1-1,..., Xnl-1, and let Fn1,..., Fnm be independent random matrices of the same form as Fn0 . We show that as n → ∞, the matrices Fn0 and m−l+1/2Fn1 +...+ Fnm have the same limiting eigenvalue distribution. To obtain our results, we apply the general framework recently introduced in Götze, Kösters, and Tikhomirov 2015 to sums of products of independent random matrices and their inverses.We establish the universality of the limiting singular value and eigenvalue distributions, and we provide a closer description of the limiting distributions in terms of free probability theory.

scholarly journals ON THE UNIVERSALITY OF THE DISTRIBUTION OF THE GENERALIZED EIGENVALUES OF A PENCIL OF HANKEL RANDOM MATRICES

2013 ◽  
Vol 02 (01) ◽  
pp. 1250014 ◽  
Author(s):  
PIERO BARONE
Keyword(s):  
Asymptotic Behavior ◽  
Integral Representation ◽  
Random Matrices ◽  
Interpolation Problem ◽  
Stationary Process ◽  
Hankel Matrices ◽  
Spherical Distribution ◽  
Generalized Eigenvalues ◽  
Specific Distribution ◽  
Finite Set

Universality properties of the distribution of the generalized eigenvalues of a pencil of random Hankel matrices, arising in the solution of the exponential interpolation problem of a complex discrete stationary process, are proved under the assumption that every finite set of random variables of the process have a multivariate spherical distribution. An integral representation of the condensed density of the generalized eigenvalues is also derived. The asymptotic behavior of this function turns out to depend only on stationarity and not on the specific distribution of the process.

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