Transformations of polynomial ensembles

0.22 MB·English

by  Arno B. J. Kuijlaars

5
1
0 Transformations of polynomial ensembles
2

n Arno B. J. Kuijlaars
a
J

0 Dedicated toEd Saff on the occasion of his 70th birthday
3

] Abstract. A polynomial ensemble is a probability density function for the
R position of n real particles of the form Z1n Qj<k(xk−xj)det[fk(xj)]nj,k=1,
P forcertainfunctionsf1,...,fn. Suchensemblesappearfrequentlyasthejoint
. eigenvalue density of random matrices. We present a number of transforma-
h
tionsthatpreservethestructureofapolynomialensemble. Thesetransforma-
t
a tions include the restriction of a Hermitianmatrix by removing one row and
m one column, a rank-one modification of a Hermitian matrix, and the exten-
sionofaHermitianmatrixbyaddinganextrarowandcolumnwithcomplex
[
Gaussians.

2
v
6
0
5 1. Polynomial ensembles
5
0 A polynomial ensemble is a probability density function on Rn of the form
.
1 1
n
0 (1.1) P(x1,...,xn)= Z ∆n(x) det[fk(xj)]j,k=1,
5 n
1 where f ,...,f is a given sequence of real-valued functions,
1 n
:
v
Xi ∆n(x)= (xk−xj)=det[xjk−1]nj,k=1, x=(x1,...,xn)∈Rn,
j<k
Y
r
a denotes the Vandermonde determinant, and Z is a normalization constant. Cer-
n
tain conditions on the functions f ,...,f have to be satisfied to ensure that (1.1)
1 n
is indeed a probability density. For example, the functions should be linearly inde-
pendent and the integrals
∞
xj−1f (x)dx, j,k =1,...,n
k
Z−∞
should be convergent. In addition, (1.1) has to be non-negative for all possible
choicesof(x ,...,x Rn. Theprobabilitydensity(1.1)onlydependsonthelinear
1 n
∈
spanofthefunctionsf ,...,f ,asonecanseebyapplyingcolumntransformations
1 n
to the second determinant in (1.1).
Aspecialcasearisesif f (x)=xk−1w(x), fork =1,...n,with w anintegrable
k
non-negative function on R such that the moments up to order 2n 2 exist. In
−
1
2 ARNOB.J.KUIJLAARS
that case (1.1) can be written as
n
1
(1.2) (x ,...,x )= ∆2(x) w(x ), x=(x ,...,x ) Rn,
P 1 n Z n j 1 n ∈
n
j=1
Y
and (1.2) is known as an orthogonal polynomial ensemble [18], as the analysis of
(1.2) relies on the polynomials that are orthogonalwith respect to w. The ensem-
bles (1.2) arise as the joint probability density of eigenvalues of unitary invariant
ensembles of Hermitian random matrices [9]. The polynomial ensembles (1.1) also
include the multiple orthogonal polynomials ensembles, see [19], where also more
examples from random matrix theory are given.
Ontheotherhand,wehavethat(1.1)isaspecialcaseofthemoregeneralclass
of biorthogonal ensembles, see [7, 8],
1
n n
(1.3) (x ,...,x )= det[g (x )] det[f (x )] ,
P 1 n Z k j j,k=1 k j j,k=1
n
which involves two sequences g ,...,g and f ,...,f of given functions. It is
1 n 1 n
known that (1.3) is determinantal [7], which means that there exists a kernel K :
n
R R R such that
× →
1
(x ,...,x )= det[K (x ,x )]n
P 1 n n! n j k j,k=1
and such that for every m=1,...,n 1,
−
(n m)!
m
(x ,...,x )dx dx = − det[K (x ,x )] .
ZRn−mP 1 n m+1··· n n! n j k j,k=1
The correlation kernel K has the form
n
n
K (x,y)= ψ (x)φ (y)
n k k
k=1
X
where span ψ ,...,ψ = span g ,...,g , span φ ,...,φ = span f ,...,f
1 n 1 n 1 n 1 n
{ } { } { } { }
and
∞
ψ (x)φ (x)dx=δ j,k =1,...,n.
j k j,k
Z−∞
In the case (1.1) we may write
n−1
K (x,y)= P (x)Q (y)
n j j
j=0
X
where P is a monic polynomial of degree j for j =0,...,n 1, the dual functions
j
−
Q ,...,Q areinthe linearspanoff ,...,f ,andthebiorthogonalitycondition
0 n−1 1 n
∞
(1.4) P (x)Q (x)dx=δ , j,k =0,...,n 1
j k j,k
−
Z−∞
is satisfied. In this case we can also consider the monic polynomial P of degree n
n
such that (1.4) also holds for j =n. This polynomial is given by
n
P (x)=E (x x )
n j
 − 
j=1
Y
 
TRANSFORMATIONS OF POLYNOMIAL ENSEMBLES 3
where the averaging is over (x ,...,x ) in the polynomial ensemble (1.1). If the
1 n
pointsx ,...,x comefromeigenvaluesofarandommatrix,thenP istheaverage
1 n n
characteristic polynomial.
2. Known transformations
Thispaperdiscussesanumberoftransformationsthatpreservethestructureof
a polynomial ensemble. These transformations come from random matrix theory,
andthetypicalsettingisthefollowing. WeassumethatX isarandommatrixwhose
eigenvalues (or squared singular values) are distributed according to a polynomial
ensemble(1.1). Thenwe performacertaintransformationto obtainfromX anew
randommatrixY,andtheresultisthattheeigenvalues(orsquaredsingularvalues)
of Y are again a polynomial ensemble.
2.1. Product with Ginibre matrix. The first example of such a transfor-
mation comes from recent work of the author with Dries Stivigny [20]. It deals
with the squared singular values of rectangular matrices. Recall that the squared
singular values of a rectangular complex matrix X are the eigenvalues of X∗X.
The transformation on X is multiplication by a complex Ginibre matrix, where a
complexGinibrematrixisarandommatrixwhoseentriesareindependentstandard
complex Gaussians.
Theorem 2.1. Let n,l,ν be non-negative integers with 1 n l. Let G be an
≤ ≤
(n+ν) l complex Ginibre matrix, and let X be a random matrix of size l n,
× ×
independent of G, such that the squared singular values x ,...,x are a polynomial
1 n
ensemble (1.1) for certain functions f ,...,f defined on [0, ). Then the squared
1 n
∞
singular values y ,...,y of Y =GX are a polynomial ensemble
1 n
1
n
(2.1) ∆ (y) det[g (y )] , all y >0,
Z˜ n k j j,k=1 j
n
where
∞ y dx
(2.2) g (y)= xνe−xf , y >0.
k k
x x
Z0 (cid:16) (cid:17)
Proof. See [20], where the proof is based on ideas taken from [3, 4]. (cid:3)
Note that g in (2.2) is the Mellin convolution of x xνe−x with f .
k k
7→
Theorem 2.1 can be applied repeatedly and it follows that the multiplication
with any number of complex Ginibre matrices preserves the structure of a polyno-
mial ensemble for the squared singular values.
Theorem2.1wasinspiredbyearlierresultsbyAkemannetal.[3,4]onproducts
of random matrices. In these papers the authors considered products of complex
Ginibre matrices (that is, X is also a complex Ginibre matrix) and they obtained
the structure (2.1)–(2.2), where in this case the functions g in (2.1) are expressed
k
asMeijerG-functions. Thisresulthassincethenbeenusedin[21,22]todetermine
the large n scaling limit of the correlation kernel at the hard edge, and in [2] to
calculatetheLyaponovexponentsasthenumberofmatricesintheproducttendsto
infinity. Seealso[14, 24, 25]forotherrecentresultsonsingularvaluesofproducts
of random matrices.
4 ARNOB.J.KUIJLAARS
2.2. Product with a truncated unitary matrix. Theorem 2.1 has an ex-
tension to a product with a truncated unitary matrix. A truncation T of a matrix
U is a principal submatrix of U. We assume that U is a Haar distributed random
unitary matrix and then T is also a random matrix.
Theorem 2.2. Let n,m,l,ν be non-negative integers with n l m and
≤ ≤
m n+ν +1. Let T be an (n+ν) l truncation of a Haar distributed unitary
≥ ×
matrix U of size m m. Let X be a random matrix of size l n, independent of
× ×
U, such that the squared singular values x ,...,x of X are a polynomial ensemble
1 n
(1.1) for certain functions f ,...,f defined on [0, ). Then the squared singular
1 n
∞
values y ,...,y of Y =TX are a polynomial ensemble
1 n
1
n
(2.3) ∆ (y) det[g (y )] , all y >0,
Z˜ n k j j,k=1 j
n
where
1 y dx
(2.4) g (y)= xν(1 x)m−n−ν−1f , y >0.
k k
− x x
Z0 (cid:16) (cid:17)
Proof. See [17]. (cid:3)
Ifweletm inTheorem2.2,then√mT tendsindistributiontoacomplex
Ginibre matrix.→A∞lso (1 x)m−n−ν−1 tends to e−x as m . In this way
− m → ∞
Theorem 2.1 can be obtained as a limiting case of Theorem 2.2.
Theorems 2.1 and 2.2 can be used repeatedly and it follows that the squared
singular values of a product of any number of Ginibre matrices with any number
of truncated unitary matrices are a polynomial ensemble.
2.3. Overview of the rest of the paper. Inspired by these results we give
an overview of other transformations that preserve polynomial ensembles. The
transformations are based on known random matrix theory calculations, see [11,
15], and our aim here is to emphasize the interpretation as a transformation of
polynomial ensembles.
The first such transformation comes from matrix restrictions. Here we are
working with a Hermitian matrix X and we remove one row and one column to
obtain Y. If X is random with eigenvalues that are distributed as a polynomial
ensemble then the eigenvalues of Y are also distributed as a polynomial ensemble.
This is our first result, see Theorem 3.2. The proof relies on a fundamental result
of Baryshnikov [6], see Theorem 3.1 below.
Then we extend this to the situationwhere X is a positive semidefinite matrix
with a fixed number of zero eigenvalues. Again we find that matrix restriction
for random matrices of this type leads to a transformation result for polynomial
ensembles, see Theorem 4.2. Interestingly enough, we can make a connection with
the product with a truncated unitary matrix, as we find in this way an alternative
proof for Theorem 2.2.
In Section 5 we consider a transformation from X to Y = X +vv∗ where X
is Hermitian, and v is a column vector of independent complex Gaussian entries.
This rank-one modification is also a transformationof polynomial ensembles as we
show in Proposition 5.1. The argument is based on a result of [16].
Finally,inSection6weconsideratransformationwhereweextendtheHermit-
ian matrix X by adding an extra column v with independent complex Gaussians,
and an extra row v∗ c consisting of v∗ and a real number c that has a real
(cid:0) (cid:1)
TRANSFORMATIONS OF POLYNOMIAL ENSEMBLES 5
normal distribution. Under appropriate conditions on the variances, we again find
a transformation of polynomial ensembles, see Proposition 6.2. This is based on
[1, 12].
3. Matrix restrictions
Let X be ann n Hermitianmatrix with distinct eigenvaluesx <x < <
1 2
× ···
x . Let U be a Haar distributed unitary matrix of size n n and let Y be the
n
×
(n 1) (n 1)principalsubmatrixofUXU∗witheigenvaluesy y y .
1 2 n−1
− × − ≤ ≤···≤
With probability one we have strict interlacing of eigenvalues
(3.1) x <y <x <y < <y <x .
1 1 2 2 n−1 n
···
ThefollowingtheoremisduetoBaryshnikov(reformulationof[6,Proposition4.2]).
Theorem3.1. IfX andY areasabove,thenthe(random)eigenvaluesy ,...,y
1 n−1
of Y have the joint density
∆ (y)
n−1
(3.2) (n 1)!
− ∆ (x)
n
on the subset of Rn−1 defined by the inequalities (3.1).
The interlacing condition is expressed by the determinant
n 1 if x y
(3.3) det χ , χ = ≤
xk≤yj j,k=1 x≤y (0 otherwise,
(cid:2) (cid:3)
withy :=+ . Indeed,forallmutuallydistinctvaluesx andy ,thedeterminant
n k j
∞
in (3.3) is 1 if and only if the interlacing condition holds and it is zero otherwise.
Thedeterminantin(3.3)hasallonesinthelastrow. Wecanreduceittoan(n 1)
− ×
(n 1) determinant by subtracting the last column from every other column, and
− n−1
expandingalongthelastrow. Thisresultsinthedeterminantdet χ .
xk≤yj<xn j,k=1
It means that the density (3.2) can be written as
(cid:2) (cid:3)
∆n−1(y) n ∆n−1(y) n−1
(3.4) det χ = det χ ,
∆ (x) xk≤yj j,k=1 ∆ (x) xk≤yj<xn j,k=1
n n
(cid:2) (cid:3) y =(y ,..(cid:2).,y ) R(cid:3)n−1, y :=+ ,
1 n−1 n
∈ ∞
where (3.4) is considered as a probability density on the unordered eigenvalues of
Y,i.e.,asaprobabilitydensityonRn−1. Thisaccountsforthedisappearanceofthe
factor (n 1)! from (3.2). Note that (3.4) is a polynomial ensemble with functions
−
y χ , for k =1,2,...,n 1.
7→ xk≤y<xn −
LetusnowassumethatX israndom,independentofU,andthatthe eigenval-
ues of X are a polynomial ensemble. Then the eigenvalues of Y are again a poly-
nomialensemble. Forthis it is important thatthe normalizationconstant 1 in
∆n(x)
(3.4) depends on X via the Vandermonde determinant ∆ (x) in the denominator.
n
We also need the Andreief identity, see [10, Chapter 3],
n n
(3.5) det[φ (x )] det[ψ (x )] dµ(x ) dµ(x )
k j j,k=1 k j j,k=1 1 ··· n
ZXn
n
=n!det φ (x)ψ (x)dµ(x)
j k
(cid:20)ZX (cid:21)j,k=1
6 ARNOB.J.KUIJLAARS
whereµis ameasureonaspace , n N, andφ ,...,φ , ψ ,...,ψ arearbitrary
1 n 1 n
X ∈
functions such that the integrals converge. The result is the following.
Theorem 3.2. Suppose that X is a random n n Hermitian matrix whose
×
eigenvalues are a polynomial ensemble (1.1) with certain functions f ,...,f . Let
1 n
Y be the principal submatrix of UXU∗ of size (n 1) (n 1), where U is a Haar
− × −
distributed unitary matrix, independent of X. Then the eigenvalues y ,...,y of
1 n−1
Y are a polynomial ensemble
1
(3.6) ∆ (y) det[g (y )]n−1 ,
Z˜ n−1 k j j,k=1
n−1
with
y
(3.7) g (y)= f˜(x)dx, k =1,...,n 1,
k k
−
Z−∞
where f˜,...,f˜ are a basis for the vector space
1 n−1
n ∞
(3.8) f = c f c ,...,c R, f(x)dx=0 .
j j 1 n
{ | ∈ }
j=1 Z−∞
X
Proof. From (3.4) it follows after averaging over the polynomial ensemble
(1.1) that the eigenvalues of Y have joint density
1 n n
∆ (y) det χ det[f (x )] dx dx
n!Zn n−1 ZRn xk≤yj j,k=1 k j j,k=1 1··· n
with y :=+ . Because of (3.5(cid:2)) we fin(cid:3)d that this is
n
∞
1 yj n
(3.9) ∆ (y)det f (x)dx .
n−1 k
Z
n (cid:20)Z−∞ (cid:21)j,k=1
Since y = + , the last row of the second determinant in (3.9) contains the
n
constants ∞ ∞f (x)dx. Change from f ,...,f to another basis f˜,...f˜ where
−∞ k 1 n 1 n
f˜,...,f˜ belong to the subspace (3.8). Then after performing suitable column
1 n−R1
operations we obtain the density
1 yj n
∆ (y)det f˜(x)dx .
Z˜n−1 n−1 (cid:20)Z−∞ k (cid:21)j,k=1
Then ∞ f˜ (x)dx=0, since otherwise the full last row in the determinant would
−∞ n 6
be zero. By expanding the determinant along the last row we obtain (3.6) with
R
functions (3.7) and a possibly different constant Z˜ . (cid:3)
n−1
An analogous result holds for singular values.
Corollary 3.3. Suppose that X is random (n+ν) n matrix whose squared
×
singular values are a polynomial ensemble (1.1). Let Y be the (n+ν) (n 1)
× −
principal submatrixofXU whereU is Haar distributedunitarymatrix, independent
ofX. Then thesquaredsingular valuesy ,...,y ofY areapolynomial ensemble
1 n−1
1
(3.10) ∆ (y) det[g (y )]n−1 , all y >0,
Z˜ n−1 k j j,k=1 j
n−1
with
y
(3.11) g (y)= f˜(x)dx, k =1,...,n 1
k k
−
Z0
TRANSFORMATIONS OF POLYNOMIAL ENSEMBLES 7
where f˜,...,f˜ are a basis for the vector space
1 n−1
n ∞
(3.12) f = c f c ,...,c R, f(x)dx=0 .
j j 1 n
{ | ∈ }
j=1 Z0
X
Proof. We can apply Theorem 3.2 since Y∗Y is the principal submatrix of
size(n 1) (n 1)ofX∗X. The integrationin(3.11)and(3.12)startsat0 since
the fun−ction×s are−defined for x 0 only. (cid:3)
≥
4. Restrictions of positive semidefinite matrices
ThefollowingisavariationonTheorem3.1. Itcanalsobeobtainedasaspecial
case of [15, Corollary 1].
Proposition 4.1. Let m n+1 and let X be an m m positive semidefinite
≥ ×
Hermitian matrix with n simple non-zero eigenvalues 0< x < x < <x and
1 2 n
···
an eigenvalue 0 of multiplicity m n 1. Let Y be the (m 1) (m 1) principal
− ≥ − × −
submatrix of UXU∗ where U is a Haar distributed unitary matrix of size m m.
×
Then with probability one, Y has exactly n non-zero eigenvalues 0 < y < y <
1 2
<y that satisfy the inequalities
n
···
(4.1) 0<y <x <y <x < <x <y <x ,
1 1 2 2 n−1 n n
···
and these non-zero eigenvalues have the joint density
(m 1)! n ym−n−1∆ (y)
(4.2) − k n
(m−n−1)! k=1 xmk−n ∆n(x)
Y
restricted to the subset of y =(y ,...,y ) Rn defined by the inequalities (4.1).
1 n
∈
Proof. For m = n + 1 this follows immediately from Theorem 3.1 and so
we assume in the proof that m n+2. We approximate X by a matrix A with
≥
eigenvalues a < <a <x < <x with a ’s close to zero. Let B be the
1 m−n 1 n j
··· ···
principalsubmatrixof UAU∗ of size (m 1) (m 1), which withprobability one
− × −
has distinct eigenvalues b < <b <y < <y that interlace with the
1 m−n−1 1 n
··· ···
eigenvalues of A. By Theorem 3.1 the joint density of these eigenvalues is
(4.3) (m 1)!∆m−n−1(b) j,k(yk−bj)∆n(y)
− ∆ (a) (x a ) ∆ (x)
m−n Qj,k k− j n
on the subset of Rm given by the interQlacing relations. The induced density on
y ,...,y is
1 n
(4.4) (m 1)! a2db am−n db ∆m−n−1(b) j,k(yk−bj) ∆n(y).
1 m−n−1
− Za1 ···Zam−n−1 ∆m−n(a) Qj,k(xk−aj) !∆n(x)
In the limit where all aj 0, j = 1,...,m n, weQalso have bj 0, j =
→ − →
1,...,m n 1. Then the factors (y b ) and (x a ) in(4.4) tend to
− − j,k k− j j,k k− j
ym−n−1 and xm−n, respectively. The resulting m n 1 fold integral can
k k k k Q Q − −
be evaluated as
Q Q
a2 am−n ∆ (b) 1
m−n−1
(4.5) db ... db =
1 m−n−1
∆ (a) (m n 1)!
Za1 Zam−n−1 m−n − −
and this does not depend on a ,...a . The result is the joint density (4.2) for
1 m−n
the non-zero eigenvalues of Y. (cid:3)
8 ARNOB.J.KUIJLAARS
The inequalities (4.1) are encoded by the determinant
n
det χ
0<yj<xk j,k=1
which for strictly increasing y < y(cid:2) < <(cid:3)y is 1 if the interlacing (4.1) holds
1 2 n
···
and 0 otherwise. Then (4.2) can be alternatively written as
(4.6) (m−1)! n ykm−n−1∆n(y) det χ n
n!(m−n−1)! k=1 xmk−n ∆n(x) 0<yj<xk j,k=1
Y (cid:2) (cid:3)
which is now considered as a density on [0, )n for unordered eigenvalues. Note
∞
that (4.6) is a polynomial ensemble on [0, ) with functions y ym−n−1χ
∞ 7→ 0<y<xk
for k =1,...,n.
Theorem 4.2. Letn m 1andν m n 1bepositiveintegers. LetX bea
≤ − ≤ − −
random positive semidefinite Hermitian matrix of sizem mwith azeroeigenvalue
×
of multiplicity m n 1 and non-zero eigenvalues x ,...,x that are a polynomial
1 n
− ≥
ensemble (1.1) for certain functions f ,...,f on [0, ). Let Y be the principal
1 n
∞
submatrix of UXU∗ of size (n+ν) (n+ν), where U is a Haar distributed unitary
×
matrix, independent of X. Then, with probability one, Y has exactly n non-zero
eigenvalues y ,...,y , and these non-zero eigenvalues are a polynomial ensemble
1 n
1
n
(4.7) ∆ (y) det[g (y )] all y >0,
Z˜ n k j j,k=1 j
n
where
1 y dx
(4.8) g (y)= xν(1 x)m−n−ν−1f .
k k
− x x
Z0 (cid:16) (cid:17)
Proof. We first assume that ν =m n 1. Then Y is obtained fromUXU∗
− −
byremovingonerowandcolumnandwecanapplyProposition4.1andinparticular
its reformulation in (4.6). Averaging (4.6) over the polynomial ensemble (1.1) we
obtain the joint density
1 ∆ (y) n ykm−n−1 det[f (x )]n det χ n dx dx ,
Z˜n n Z[0,∞)nk=1 xmk−n k j j,k=1 0<yj<xk j,k=1 1··· n
Y (cid:2) (cid:3)
for a certain constant Z˜ (which also depends on m). By the Andreief identity
n
(3.5) this leads to
1 ∞ ym−n−1
j
∆ (y)det f (x)χ dx
Z˜n n "Z0 xm−n k 0<yj<x #
with a new constant Z˜ . This is a polynomial ensemble (4.7) with functions
n
∞ ym−n−1
g (y)= f (x)χ dx
k xm−n k 0<y<x
Z0
∞ ym−n−1
(4.9) = f (x)dx, y >0.
xm−n k
Zy
The substitution x y in (4.9) leads to the expression (4.8) with ν =m n 1.
7→ x − −
This is the Mellin convolution of f with the function χ where we define
k m−n−1
χ :[0, ) R: x xkχ .
k 0<x<1
∞ → 7→
TRANSFORMATIONS OF POLYNOMIAL ENSEMBLES 9
Thus g = χ f , if ν = m n 1, where is used here for the Mellin
k m−n−1 k
∗ − − ∗
convolution
∞ y dx
(f g)(y)= f(x)g .
∗ x x
Z0 (cid:16) (cid:17)
For general ν m n 1 we can use the above argument repeatedly, and
≤ − −
we find a polynomial ensemble (4.7) with functions g that are iterated Mellin
k
convolutions of the functions f , namely
k
g =χ χ χ f , k =1,...,n.
k ν ν+1 m−n−1 k
∗ ∗···∗ ∗
It is easy to calculate that
(χ χ χ )(x)=xν(1 x)m−n−ν−1, 0<x<1,
ν ν+1 m−n−1
∗ ∗···∗ −
and thus we obtain the formula (4.8) for the functions g . (cid:3)
k
Anattentivereadermayhavenoticedthattheformulaforg in(4.8)coincides
k
with the one appearing in (2.4) in Theorem 2.2. This is no coincidence since we
can use Theorem 4.2 to give an alternative proof of Theorem 2.2.
Proof of Theorem 2.2. Let X be an l n matrix, and put
×
XX∗ 0
X˜ =
0 0
(cid:18) (cid:19)
whichis anm mmatrixwithm l rowsandcolumns containingonly zeros. It is
clear that the×squaredsingular val−ues of X are equal to non-zero eigenvalues of X˜.
Also if U is a unitary matrix of size m m and T is its left upper block of size
×
(n+ν) l then
× TXX∗T∗
UX˜U∗ = ∗
(cid:18) ∗ ∗(cid:19)
where denotes a certain unspecified entry, whose value is not important for us.
In othe∗r words, TXX∗T∗ is equal to the principal submatrix of UX˜U∗ of size
(n+ν) (n+ν). Assuming U is Haar distributed over the unitary group, and
×
the squared singular values of X are a polynomial ensemble (1.1), independent
of U, we then find from Theorem 4.2 that the non-zero eigenvalues of TXX∗T∗
are a polynomial ensemble (2.1) with functions (4.8). The non-zero eigenvalues
of TXX∗T∗ are the same as the squared singular values of TX and Theorem 2.2
follows. (cid:3)
5. Rank one modification
Let X be a Hermitian n n matrix with eigenvalues x < x < < x .
1 2 n
× ···
We take Y = X + vv∗ where v is a vector of length n. Then the eigenvalues
y of Y interlace with those of X, as follows from the Courant-Fischer Theorem,
j
see e.g. [23, chapter 7.5]. We let v = (v ,...,v )t be a vector of independent
1 n
complex random variables whose real and imaginary parts are independent and
have a N(0,1/2) distribution. Then the distribution of the eigenvalues of Y is
given in [16, Appendix E] as
n
∆ (y)
(5.1) e−(yj−xj) n
∆ (x)
n
j=1
Y
on the subset of Rn given by the interlacing conditions
(5.2) x <y <x < <x <y .
1 1 2 n n
···
10 ARNOB.J.KUIJLAARS
The following result is an immediate consequence.
Proposition 5.1. Let Rev , Imv , for j = 1,...,n be mutually independent
j j
normal random variables with mean zero and variance 1/2. Let X be a random
Hermitian matrix of size n n, independent of v = (v ,...,v )t, whose eigenval-
1 n
×
ues are a polynomial ensemble (1.1) with certain functions f ,...,f . Then the
1 n
eigenvalues y ,...,y of Y =X+vv∗ are a polynomial ensemble
1 n
1
(5.3) ∆ (y) det[g (y )]n ,
Z˜ n k j j,k=1
n
where
∞
(5.4) g (y)= e−xf (y x)dx, k =1,...,n.
k k
−
Z0
Thus g is the convolution of f with x e−xχ .
k k x≥0
7→
Proof. The interlacing (5.2) is encoded by a determinant, and it follows that
(5.1) is a polynomial ensemble
n
(5.5) n1! e−(yj−xj)∆∆n((xy))det χxk<yj nj,k=1,
n
j=1
Y (cid:2) (cid:3)
where now we disregardthe orderingof the y ’s and consider (5.5) as a probability
j
density on Rn.
Weaverageoverx ,...,x distributedasin(1.1). ByAndreief’sidentity(3.5),
1 n
we obtain for the density of the eigenvalues of Y
1 yj n
∆ (y)det e−(yj−x)f (x)dx .
Z˜n n (cid:20)Z−∞ k (cid:21)j,k=1
Changingvariablesx y xintheintegralinthedeterminant,wearriveat(5.3)
j
with functions (5.4). 7→ − (cid:3)
Here is a variation on the same theme.
Proposition 5.2. Let n,ν 1. Let Rev , Imv , for j = 1,...,n + ν be
j j
≥
mutually independent normal random variables with mean zero and variance 1/2.
Let X be a random (n+ν) (n+ν) positive semidefinite Hermitian matrix, in-
×
dependent of v ,...,v , with exactly n positive eigenvalues x ,...,x that are a
1 n 1 n
polynomial ensemble (1.1) with certain functions f ,...,f on [0, ). Then, al-
1 n
∞
most surely, Y = X +vv∗ has an eigenvalue zero of multiplicity ν 1 and n+1
−
positive eigenvalues y ,...,y that are a polynomial ensemble
1 n+1
1
(5.6) ∆ (y) det[g (y )]n+1 , all y >0,
Z˜ n+1 k j j,k=1 j
n+1
with functions
g (y)=yν−1e−y,
1
(5.7) y
g (y)=yν−1e−y x−νexf (x)dx, k =1,...,n,
k+1 k
Zc
for y >0, where c (0, ) is an arbitrary but fixed positive real number.
∈ ∞
We may also take c = 0 or c = in (5.7) provided that the integrals are all
∞
convergent.