On the joint eigenvalue distribution for the matrix ensembles with non zero mean

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Pram-ana, Vol. 15, No. 1, July 1980, pp. 45-51. © Printed in India.

On the joint eigenvalue distribution for the matrix ensembles with

n o n z e r o m e a n

N A Z A K A T U L L A H

Tara Institute of Fundamental Research Homi Bhabha Road, Bombay 400 005, India MS received 6 September 1979

Abstract. Exact distributions are given for the two-dimensional case when the mean of the off-diagonal element is non-zero. The joint eigenvalue distribution for the N dimensional case, derived using the volume element in the space of N × N orthogonal matrices, is checked by rederiving the exact results for N=2. The smooth nature of the N-dimensional joint distribution supports the claim of the method o f moments that the single eigenvalue distribution is a smooth function of the ratio of mean-to- mean square deviation.

Keywords. Randon matrices; new matrix ensembles.

1. Introduction

T h e r a n d o m m a t r i x ensembles have been successfully used in the p a s t ( M e h t a 1967) to study the average properties o f the excited states in c o m p o u n d nucleus. Because o f the r a n d o m sign rule f o r the off-diagonal elements, the m e a n value o f the off- diagonal elements was always t a k e n to be zero in these studies. L a t e r ( N a z a k a t UUah et al 1970) it was realised t h a t for a satisfactory description o f the nuclear configuration interaction p r o b l e m new ensembles have to be i n t r o d u c e d in which the m e a n value o f the off-diagonal elements is n o t zero. Because o f the c o m p l e x i t y o f the p r o b l e m a perturbative t r e a t m e n t was carried out. Recently (Edwards a n d Jones 1976; K o t a a n d P o t b h a r e 1977) there has been considerable interest in the study o f single eigenvalue distribution o f a r a n d o m matrix ensemble each e l e m e n t o f which has a G a u s s i a n distribution a n d non-zero mean. T h e results o b t a i n e d b y I ( o t a a n d P o t b h a r e (1977) using the m e t h o d o f m o m e n t s are very different t h a n the ones given by E d w a r d s and Jones (1976) using what is k n o w n as n ~ 0 trick. T h e m a i n result which one is interested in is to see whether the single eigenvalue distribution is a s m o o t h function for all values o f the ratio o f m e a n - t o - m e a n square deviation or has some kind o f discontinuity in it. In a private c o m m u n i c a t i o n P a r i k h (1979) has also expressed the opinion that the method o f m o m e n t s a r g u m e n t leads one to believe t h a t the single eigenvalue has a smooth distribution. T h e p u r p o s e o f the present w o r k is to give an exact joint eigenvalue distribution a n d see w h a t c a n be predicted a b o u t the nature o f single eigenvalue distribution f r o m it.

We shall first derive the j o i n t distribution for the two-dimensional cases in the next section. This will serve the purpose o f checking the general derivation which is given in §3. I n §4 we apply general f o r m a l i s m to the three dimensional case. C o n - cluding r e m a r k s are presented in §5.

45

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2. Joint distribution in two dimensions

Let us consider a real symmetric 2 × 2 matrix having the following distribution 2

t~=l

where A is the mean value and o is the variance of the off-diagonal element. We have used a slightly different distribution of Ht, u than the one used by Edwards and Jones (1976). In our distribution the variance of the diagonal element is twice that of the off-diagonal element. We have done this in order that our distribution goes over to the earlier distribution (Mehta 1967) when A ~0 for all values of the dimension of the matrix H.

The mean value A of all the matrix elements is taken to be the same (Edwards and Jones 1976; Nazakat Ullah et a l 1970). This is the simplest description of the actual physical situation and is almost the same what one does in the mean-field theory. In the many-nucleon system further support for this simplifying aspect of the problem comes from the diagonalisation of large shell-model matrices.

In order to get the joint eigenvalue distribution directly from the distribution of matrix elements of H, we introduce the following two variables

u --//11 + Hn, (2a)

v = H n H a 2 - - H~z. (2b)

According to the theory of probability (Kendall 1945) the unnormalised joint distribution of u, v using the distribution (1) is given by

P ( u , v) = K f S [ u - - ( H n + H22)] 8 [ 0 - (H n H2z -- H~J]

2

/~---1 Carrying out the integrations, we get

( )

P ( u , v ) : K e x p - - - - 1 tu 2 _ 2 a u _ 2 v ] I o ~ / ~ - - 4v

(4)

where K now denotes the appropriate normalisation constant.

From the theory of matrices we know that the variable u and v are related to the eigenvalues El, Es in the following way:

u = E~ + E,, (Sa).

v = E~E~,.

(5b)

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Joint eigenvalue distribution for matrix ensembles

47 Using expressions (4), (5) and the Jacobian of the transformation (Kendall 1945) from the variables u, v to E 1, E~ we obtain the following expression for the j o i n t eigenvalue distribution o f H

P(E 1, E2) = K

( I [ E ~ . - - [ - E ~ - - 2 ~ ( E 1 - L E 2 ) ] ] E 1 - - E 2 , I o ( 5 ( E I - - E 2 ) ) )

^{( 6 )}

exp - - ~ , ,

where K is fixed by the normalisation condition

o0 o0

f f PeEl, E,~)dE 1 dE 2 = 1, --00 --00

and I 0 is the modified Bessel function (Abramowitz a n d Stegun 1965).

Comparing the distribution (6) with the earlier distribution when A--0, we find that a non-zero mean value for all the elements of H has given rise to the additional factor 10 apart from shifting each eigenvalue by A.

From the joint distribution (6) it is easy to see that the spacing distribution

P(S), S : ]E1--E ~

] for the non-zero mean ensemble is given by

P(S) = (4a2)-X [exp (-- ~-~o2)] S exp (-- ~-~) lo(~---~S) • (7)

Next integrating out one o f the eigenvalues in (5) we get the following expression for the distribution o f single eigenvalue

The integral

oO

P(E) -- K

exp ( --

0

o'o

o

(8)

(9)

in expression (8) can be integrated in the form of the series using the expansion o f I o G -- ~ o o ( 2 k + l ) [ ()~/2o~)~(4o~)~+x exp (~ (E--A)2 /

k=O (k!) Z 2 ~+~ " ~ /

E--A _

v + v 2 k + g , ' (lo)

(4)

where U are the functions related to parabolic cylinder functions and tabulated by Abramowitz and Stegun (1965).

3. General distribution

It is obvious that the method of §2 cannot be generalised easily to N dimensions.

We now give a prescription for finding the N-dimensional joint distribution.

Let us consider a real symmetric N x N matrix H having the following distribution

N N

P ( { H t t ~ } ) : = K e x p I - - ~ - ~ ~ [ ~ 1 (/-I~t,--A)2-~-2 ~ (Ht~'-A)2]} ' t~=l p < a = l

(11) The matrix H is related to the eigenvalues E k and orthogonal matrix T by the following relation

H : "TE T.

The eigenvalue distribution is obtained by integrating out over T.

(12) this is given by

(12) Using (11) and

[ 1 Z ]

P ( { E t , } ) - : K exp -- ~ (E~ -- 2 ;9 Eta) II ] E~, -- Ev [ p<v

/~=1

f exp [~ek ( ~ r,~ rkv) ] at,

k p<v

(13)

I I I Et~ - - Ev I arises from the Jacobian of the transformation (12) and /t<v

(15)

(16) The factor

d T denotes the volume element in the space of N x N orthogonal matrices.

functions (Nazakat Ullah 1964) this can be written as .

J i ]<k i

Equivalently this can also be written as

Using (14) the integral in (13) can be written as

f exp ~: ( ~ - i (& -eu) ~ T,, r,,)dr.

i<y

(14) Using 8

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Joint eigenvalue distribution for matrix ensembles

⁴⁹

Now since

i<j i

Using (15), (16) can be written as N - - I

ox~(_¢) [ X ,-',*~] f**

N - - I A

exP2-~ s E

(Ek--EN)

k=~l k = l

We now apply an orthogonal transformation to T

T' =R T,

⁽¹⁸⁾

and choose the elements in the first row of R as

1/.v/N.

This gives

Z~ Tk~='V/~T~I

and since under an orthogonal transformation the volume element (14) or (15) re- mains invariant, expression (17) can be rewritten as

N--1 N - - I

e x p [ - - ~ ( E

Et'--(N--1)EN)3exp[A~ ~ (Et'--EN'T~'~] dT"

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k = l k = l

Since in the exponent only the elements o f the first column appear we can formally integrate o u t over the remaining N--1 columns and get the following expression for the joint probability o f the eigenvalues

P ( ( e ~ ) ) = K exp - - ~ = 1 i, < ,

N--1

k = l N--1

E (~,-e,,) 7a a(~ r:~-l) u a~;,. (2o)

k--1 k k

Integrating over T~/1 we can also write the joint distribution as N

P ({e~,)) : K exp - - ~ P" < .

(21)

N--1

oxp _~,( ~ _~_1,,~ /**

k = l

N--1 N--1

k=l k=l

P.---4

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where the integration is to be carried out over the region N--1

k = l

and K is the appropriate normalisation constant. Expression (21) is the desired expression for the joint distribution of E~. We see that the presence of ~ gives rise to an additional factor which is fairly complicated. But free from any discontinuity for all values of ~/~. Further integration over the remaining variables of T is not easy. However for N = 3 , it can be done fairly easily. This is done in the next section. We also note from expression (21) that to get the single eigenvalue distribution the integration over N--1 eigenvalues is also quite involved for general N and can be done explicitly for small values of N only. However as mentioned in §1 our main interest was to see whether the single eigenvalue distribution is a smooth function of ~/a or has some discontinuity for some value of ~/~. It is fairly obvious from the structure of expression (21) and also the explicit integrations for N = 2 , 3 that further integration over N--1 eigenvalues cannot give rise to any discontinuity in the single eigenvalue distribution.

4. T h r e e - d i m e n s i o n a l distribution

Before we work out the three-dimensional distribution we would like to show that expression (21) reproduces the result of §2 when

N=2.

For N = 2 we get from expression (21)

2

P (Ex" E~) =Kexp [ -14cr2 ~ (E~--2A Et~)J ] Ex-E~ [

1

e x p [ - - ~ z ( E x - - E z ) ]

f exp (A/°~' (Ex--E2) TI~ (1--TI~)-I/$ dZtll . --I

Carrying out the integration over

dT~x

we get the same expression as (5). This provides a check on expression (21).

For the three-dimensional case let us write

A = e x p [ - - ~

(EI+Ez--2Ez) ] f

exp3~2~' [(Ex--Es)T;: + (Ez--F-~)2rX]

d T;'x d T~I .

(22)

a/1 - - (T~I + T~I) ,~ ,2

Introducing polar co-ordinates Tit = p cos 0, T6x = e sin 0, we get

1 2n

A = f f exp (3ApZ/4o ~)

(EI+E~--2Es)

exp (3M'/4o ~)

(EI--Ez) cos 20

p = 0 o = 0 pdp de.

a/1 - - e 2

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Joint eigenvalue distribution for matrix ensembles

51 Carrying o u t the integrations o v e r g a n d 0 we get,

A = . n "s/~ exp - -

(Ex+Ez--2E3)

m ! (n!) 2 2 an+l I" ( m + 2 n + ~ )

m s I'1

[(3;~/4~ 2)

(EI+Ea--2Es) ]"

[(3)t/4tr 2) (El--E2)] ~". (23) C o m b i n i n g expressions (21) a n d (23) the three-dimensional distribution is given b y

3

[ A ~ (E~--2AE,)] H 1 E , - - E . I A.

(24, P ( ( G ) ) : K exp - - ~ A t* < ~, = 1

t.,=l

5. Concluding remarks

Exact analytical distributions o f various quantities have b e e n derived f o r N---2 a n d f o r non-zero m e a n o f the off-diagonal element. A n exact j o i n t distribution f o r N dimensions is derived using 8-function technique. Its validity is checked b y rederiving the t w o - d i m e n s i o n a l distribution f r o m it. L o o k i n g at the N - d i m e n s i o n a l j o i n t distribution which is written in the f o r m o f integral o v e r a n N - d i m e n s i o n a l unit vector, we see t h a t it does n o t show any discontinuity f o r a n y value o f the ratio o f m e a n - t o - m e a n square deviation. We therefore conclude t h a t the single eigenvalue distribution, will be a s m o o t h distribution, thus supporting the result which is also arrived at b y the a r g u m e n t based on the m e t h o d of m o m e n t s .

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Kendall M G 1945

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(London: Charles Griffin) Vol. 1 Kota V K B and Potbhare V 1977

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(New York: Academic Press)

Nazakat Ullah 1964

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58 65

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Can. J. Phys. 48

842 Parikh J C 1979 Private Communication

On the joint eigenvalue distribution for the matrix ensembles with non zero mean