ON HADAMARD POWERS OF RANDOM WISHART MATRICES JNANESHWAR BASLINGKER ABSTRACT . A famous result of Horn and Fitzgerald is that the -th Hadamard power of any nn

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ON HADAMARD POWERS OF RANDOM WISHART MATRICES

JNANESHWAR BASLINGKER

ABSTRACT. A famous result of Horn and Fitzgerald is that the β-th Hadamard power of any n×n

positive semi-deﬁnite (p.s.d) matrix with non-negative entries is p.s.d ∀β≥n−2and is not necessar-

liy p.s.d for β < n −2,with β /∈N. In this article, we study this question for random Wishart matrix

An:= XnXT

n, where Xnis n×nmatrix with i.i.d. Gaussians. It is shown that applying x→ |x|α

entrywise to An, the resulting matrix is p.s.d, with high probability, for α > 1and is not p.s.d, with

high probability, for α < 1. It is also shown that if Xnare bnsc × nmatrices, for any s < 1, the

transition of positivity occurs at the exponent α=s.

1. INTRODUCTION

Entrywise exponents of matrices preserving positive semi-deﬁniteness has been a topic of active

research. An important theorem in this ﬁeld is the result of Horn and Fitzgerald [3]. Let P+

ndenote

the set of n×np.s.d. matrices with non-negative entries. Schur product theorem gives us that

the m-th Hadamard power A◦m:= [am

ij ]of any p.s.d. matrix A= [aij]∈ P+

nis again p.s.d. for

every positive integer m. Horn and Fitzgerald proved that n−2is the ‘critical exponent’ for such

matrices, i.e., n−2is the least number for which A◦α∈ P+

nfor every A∈ P+

nand for every real

number α≥n−2. They considered the matrix A∈ P+

nwith (i, j)-th entry 1 + εij and showed

that if αis not an integer and 0< α < n −2, then A◦αis not positive semi-deﬁnite for a sufﬁciently

small positive number ε(see [6]).

We consider a random matrix version of this problem. Let X:= [Xij]be a n×nmatrix, where

Xij are i.i.d standard normal random variables. Deﬁne An:= XXT

nand |An|◦αas the matrix

obtained by applying x→ |x|αfunction entrywise to An. Let Bn,α := |An|◦α.

We are interested in the values of real α > 0for which the matrix Bn,α is positive semi-deﬁnite,

with high probability. Simulations show that for large values of n, if α > 1then with high prob-

ability, Bn,α is positive semi-deﬁnite and for α < 1, with high probability, Bn,α is not positive

semi-deﬁnite (as shown in Table 1).

We state and prove the theorem that these observations from simulations are indeed true. In

fact we prove a stronger result. Fix any s≤1and let m=bnsc. Let Xn:= [Xij ]be a m×nmatrix,

where Xij are i.i.d standard normal random variables. Deﬁne An,s := XnXT

nand Bn,α,s := |An,s|◦α.

Let λ1(A)denote the smallest eigenvalue of A. We prove the following main result.

2010 Mathematics Subject Classiﬁcation. 60B20, 60B11 .

Key words and phrases. Wishart matrices, Positive semi-deﬁnite, Hadamard power.

arXiv:2210.15320v1 [math.PR] 27 Oct 2022

Theorem 1. ∃εs>0such that for α > s, as n→ ∞

P(λ1(Bn,α,s)≥εs)→1.

For α < s, as n→ ∞

P(λ1(Bn,α,s)<0) →1.

Remark 2. Simulations show Theorem 1holds if i.i.d Gaussians are replaced by other i.i.d random

variables with ﬁnite second moment like Uniform(0,1), Exp(1) and even heavy tailed distributions

like Cauchy distribution, distributions with densities f(x) = bx−1−b,∀x≥1, all with transition of

positivity at exponent α=s. Note that in the last case one does not have ﬁnite mean if bis small.

This suggests that the transition of matrix positivity happens for a large family of distributions. In

this direction we prove the below proposition where we show that Bn,α,s is p.s.d for the range of

α > 2s, when Xnhas sub-Gaussian entries.

Proposition 3. Let the entries of Xnbe i.i.d sub-Gaussian random variables with mean 0and unit variance.

Fix α > 2sand ε > 0. Deﬁne Bn,α,s as before. Then as n→ ∞

P(λ1(Bn,α,s)≤1−ε)→0,(1)

P(λm(Bn,α,s)≥1 + ε)→0.(2)

s α λ1

1 0.98 −0.288

1 0.99 −0.246

1 1.06 0.016

1 1.07 0.046

0.8 0.78 −0.076

0.8 0.79 −0.049

0.8 0.81 0.017

0.8 0.82 0.041

TABLE 1. Table of smallest eigenvalues for varying αand swith n= 5000.

Remark 4. Although Theorem 1and Proposition 3hold for m= Θ(ns), for deﬁniteness we ﬁx

m=bnsc. For m=a×nfor ﬁxed a > 0, the transition of positivity is at exponent 1. For the

critical exponent to be less than 1, we need m= Θ(ns)with s < 1, which is much smaller, unlike

in the study of spectrum of Wishart matrices.

A standard way to study the distribution of eigenvalues of a random matrix is to look at the

limit of empirical spectral distributions using method of moments. For example, Wigner’s proof of

semi-circle law for Gaussian ensemble uses this method (For more see [1]). In our case, the entries

of the matrix Bn,α,s are sums of products of random variables and the entries on the same row or

column are correlated. The entrywise absolute fractional power makes this problem intractable, if

we try to use method of moments. As we are interested only in the existence of negative eigenval-

ues, we manage to avoid computing all the moments.

1.1. Outline of the paper:

First we prove Proposition 3in Section 2. This is done using Gershgorin’s circle theorem and the

sub-exponential Bernstein’s inequality. Note that this proposition is not needed to prove Theorem

The proof of Theorem 1is divided into two parts. In the ﬁrst part of the proof, we consider the

range α < s. Let Cn,α,s := Bn,α,s

ns−α

2. For ease of notation, we write Cn,α,s as Cm.Cmis a m×m

matrix where m=bnsc. Deﬁne the diagonal matrix Dm, with Dm(i, i) := Cm(i, i)−`α

ns/2and

Em:= Cm−Dm−`α

ns/2Jm, where `αis as deﬁned in Subsection 1.2. We use the following lemma,

whose proof is given in Section 3, to conclude that EESD of Bn,α,s has positive weight on negative

reals.

Lemma 5. Let ¯µEmbe the EESD of Em. Then

i) Limit of ﬁrst moment of ¯µEmis 0

ii) Limit of second moment of ¯µEmis a positive constant

iii) The fourth moments of ¯µEmare uniformly bounded.

Using a concentration of measure result, we show that with high probability, Bn,α,s has negative

eigenvalues. This is done in Section 3.

In the second part of the proof, we consider the range s < α. We further divide this range

by looking at k+1

ks<α, where kis an integer greater than 1and let k→ ∞. For k+1

ks <

α, we consider Cm, a modiﬁcation of Bn,α,s, whose EESD has 2k-th moment converging to 0to

conclude that the probability of Bn,α,s having a negative eigenvalue converges to 0. We then let k

be arbitrarily large. This is done in Section 4.

1.2. Notation.

1) m=bnsc.

2) λ1(A)and λm(A)denote the smallest and largest eigenvalues of Arespectively.

3) Ridenotes the ith row of Xn. (RT

i∼N(0, In)in Sections 3,4but not necessarily in Section 2).

4) ρij =hRi,Rji

kRikkRjk.

5) `α=E[|Z|α], where Zis a standard normal random variable.

6) Jn=All ones matrix of size n×nand In=n×nidentity matrix.

7) Fi,j =The sigma algebra generated from the ith row and jth row of Xn.

8) σi=kRik/√n.

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ONHADAMARDPOWERSOFRANDOMWISHARTMATRICESJNANESHWARBASLINGKERABSTRACT.AfamousresultofHornandFitzgeraldisthatthe-thHadamardpowerofanynnpositivesemi-denite(p.s.d)matrixwithnon-negativeentriesisp.s.d8n2andisnotnecessar-liyp.s.dfor

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ON HADAMARD POWERS OF RANDOM WISHART MATRICES JNANESHWAR BASLINGKER ABSTRACT . A famous result of Horn and Fitzgerald is that the -th Hadamard power of any nn

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