Matrix Polar Decomposition and Generalisations of the Blaschke-Petkantschin Formula in Integral Geometry

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by  Peter J. Forrester

MATRIX POLAR DECOMPOSITION AND GENERALISATIONS OF THE
BLASCHKE-PETKANTSCHIN FORMULA IN INTEGRAL GEOMETRY
PETERJ.FORRESTER
7
1
0 Abstract. Inthework[Bull,Austr.Math.Soc.85(2012),315-234],S.R.Moghadasi
2 hasshownhowthedecompositionoftheN-foldproductofLebesguemeasure
n onRnimpliedbymatrixpolardecompositioncanbeusedtoderivetheBlaschke-
a Petkantschindecompositionofmeasureformulafromintegralgeometry.Weuse
J
knownformulasfromrandommatrixtheorytogiveasimplifiedderivationofthe

7 decompositionofLebesgueproductmeasureimpliedbymatrixpolardecomposi-
1
tion,applyingtootothecasesofcomplexandrealquaternionentries,andwegive

correspondinggeneralisationsoftheBlaschke–Petkantschinformula.Anumberof
]
R applicationstorandommatrixtheoryandintegralgeometryaregiven,including
P tothecalculationofthemomentsofthevolumecontentoftheconvexhullof
. k≤N+1pointsinRN,CN orHN withaGaussianoruniformdistribution.
h
t
a
m
[ 1. Introduction

1
v Offundamentalimportancetomatrixtheoryisthesingularvaluedecomposi-
5 tion. Let M bean n×N realmatrix,andsuppose n(cid:62) N sothatthecolumnsof
0 M canformabasisofan N-dimensionalsubspaceinRn. LetU (V)bean n×n
5
4 (N×N)realorthogonalmatrix,andlet Λ bean n×N diagonalmatrix,diagonal
0
entries σ (j = 1,...,N) each non-negative. The singular value decomposition
. j
1
takestheform
0
7 M =UΛVT. (1)
1
v: Since MTM =VΛTΛVT oneseesthatthe{σ2}aretheeigenvaluesofMTMandV
j
Xi isthecorrespondingmatrixofeigenvectors,whilethefactthat MMT =UΛΛTUT
r tellsusthatthe{σ2}arethenon-zeroeigenvaluesof MMT,andthecolumnsofU
a j
aretheeigenvectors.
Anequivalentwaytowrite(1)istofirstnotethat
(cid:34) (cid:35)
Λ = I˜ diag(σ ,...,σ ), I˜ := IN×N .
n,N 1 N n,N
0
(n−N)×N
Thisshows
M =U diag(σ ,...,σ )VT, (2)
1 1 N
whereU =UI˜ ;thusU consistsofthefirst N columnsofU. MultiplyingU
1 n,N 1 1
in(2)byVTV —whichistheidentity—ontherightshowsthatacorollaryofthe
1
2 PETERJ.FORRESTER
singularvaluedecompositionisthepolardecomposition
M = QP, (3)
where Q =U VT hasits N columnsforminganorthonormalsetofvectorsinRn,
1
and P =Vdiag(σ ,...,σ )VT isan N×N realsymmetricmatrix.
1 N
The decompositions (2) and (3) both have analogues for M complex. The
singularvaluedecomposition(2)hastheform
M =U diag(σ ,...,σ )V†, (4)
1 1 N
whereV isan N×N unitarymatrix,andthecolumnsofU formanorthonormal
1
setof N vectorsinthecomplexvectorspaceCn. Thepolardecomposition(3)has
theform
M = ZH, (5)
where Z =U V† alsohasits N columnsforminganorthonormalsetofvectorsin
1
Cn and H =Vdiag(σ ,...,σ )V† iscomplexHermitian.
1 N
InarecentpaperS.R.Moghadasi[17]hasconsideredthedecompositionsof
Lebesguemeasureimpliedby(3)and(5). Intherealcaseanapplicationwasgiven
toprovideanewderivationofadecompositionofLebesguemeasurefromintegral
geometrydueoriginallytoBlaschke[2]andPetkantschin[20](ormoreprecisely,
anequivalentformthereof,duetoMiles[16]). Afurtherapplicationwasgivento
the computation of the moments of the determinant of an N×N standard real
Gaussianmatrix.
It is an aim of the present paper to give a different derivation to that in [17]
oftheLebesguemeasuredecompositionsimpliedby(3)and(5). Thisderivation,
tobecarriedoutinSection2,isdeducedfromknowledgeofamatrixchangeof
variablesformulacoretothestudyoftheso-calledWishartmatricesinrandom
matrixtheory,whichinturnisequivalenttoamatrixpolarintegrationtheorem. It
allowsforanaturalextensiontothecasethattheentriesof Marerealquaternions,
representedas2×2complexmatrices
(cid:34) (cid:35)
z w
. (6)
−w z
Thematrixpolarintegrationtheoremcanthen,following[17],beusedtoderive
Blashke-Petkantschintypedecompositionofmeasureformulas,nowforthereal,
complexandrealquaternioncases. ThisisdoneinSection3.
In Section 4, a matrix integration formula ofSection 2 is usedto deduce the
matrix distribution of U(M†M)1/2, for U a random unitary matrix with entries
whicharereal,complexorrealquaternion,correspondingtothoseof M,inthe
casethatthematrixdistributionof M dependson M†M. Asanotherapplication,
theevaluationofthemomentsofdeterminantsassociatedwithcertainclassesof
random matrices with entries from these number fields is given, as is an affine
MATRIXPOLARDECOMPOSITIONANDTHEBLASCHKE-PETKANTSCHINFORMULA 3
versionofthisresult. Intherealcase,theaffineversionrelatestothevolumeof
theconvexhullof k ≤ N+1pointsuniformlydistributedintheunitballinRN,
oronitssurface,asfirstcalculatedbyKingman[11]andMiles[16].
2. DecompositionofMeasureImpliedbyMatrixPolarDecomposition
Intherealcasethesingularvalueandpolardecompositionsaregivenby(2)
and(3),whileinthecomplexcasetheyaregivenby(4)and(5). Alsoofinterestis
therealquaternioncase,inwhichtheentriesof M havethe2×2blockstructure
(6). Sincethereareingeneralfourindependentrealentriesin(6),wewillannotate
thematricesin(4)and(5)toinvolvesuperscripts4sothatforthesingularvalue
decompositionwehave
M(4) =U(4)diag(σ I ,...,σ I )V(4)†, (7)
1 1 2 N 2
andforthepolardecompositionwehave
M(4) = Z(4)H(4), (8)
As a complex matrix, M(4) is of size 2n×2N. The matrix V(4) is a 2N×2N
unitarymatrixwith2×2realquaternionentries,andthusafterconjugationbya
permutationmatrixisamemberoftheclassicalgroupSp(2N). Eachblockcolumn
ofU(4) canberegardedasamemberoftherealquaternionvectorspaceHn,and
1
the N block columns together form an orthonormal set. Note that the singular
values in (7) are doubly degenerate. This is in keeping withW(4) := M(4)†M(4)
havingthesymmetryW(4) = Z˜ W(4)Z˜−1,where
2N 2N
(cid:34) (cid:35)
0 −1
Z˜ :=I ⊗ ,
2N N
1 0
which implies the eigenvalues of W(4) are doubly degenerate. In (8) Z =
4
U(4)V(4)†,andthis2n×2Nmatrixhasthesamepropertiesasthosehighlightedfor
1
U . Also H(4) = V(4)diag(σ I ,...,σ I )V(4)†, whichis a 2N×2N Hermitian
1,4 1 2 N 2
matrixwithrealquaternionblocks(6).
Introducethenotation (dM) fortheproductoftheindependentdifferentials
(real and imaginary parts) of the entries of the matrix M. Let β = 1, 2 or 4
accordingtotheentriesof Mbeingreal,complexorrealquaternion. Asafirstuse
ofthisnotation,wewriteF todenotethesenumberfields. For M asappearing
β
in(3),(5)and(8), (dM) consistsof βnN independentdifferentials.
Consider the matrix W = M†M. In the real case it follows from (5) that
W = H2; in the real quaternion case it follows from (8) that W = (H(4))2. In
mathematical statistics a change of variables between the entries of M and the
entries of W was first studied by Wishart [26] in the real case. This was further
refined by James [10] (see the text book treatment in [18]) to obtain a change of
variables formula associated with both (2) and (3). The analogue of this in the
4 PETERJ.FORRESTER
complexcaseisdetailedin[7,§3.2.3],andthemodificationrequiredforthereal
quaternion case given in [7, Ex. 3.2 q.5]. The change of variables formula in all
threecasesisconvenientlysummarisedin[4,Prop.4]andreads
(cid:16) (cid:17)
(dM) =2−N(detW)β(n−N+1)/2−1(dW) U†dU . (9)
1 1
(cid:0) (cid:1)
Here U†dU istheinvariantmeasureonthesetofn×N matriceswithelements
1 1
in F , and having the property that the N columns form an orthonormal set in
β
thevectorspaceFn. ThesematricesaresaidtodefinetheStiefelmanifold,tobe
β
denotedVβ sowehaveU ∈ Vβ and(cid:0)U†dU (cid:1)istheinvariantmeasureonthis
N,n 1 N,n 1 1
manifold. Also,intermsofthesingularvalues,wedefine
N
detW = ∏σ2 (10)
l
l=1
inallthreecases. Thustheusualdefinitionofthedeterminantismodifiedinthe
case β = 4, where W has eigenvalues {σ2} each with multiplicity 2; in (10) the
l
multiplicityisignored.
From(9)wecanreadoffamatrixpolarintegrationtheorem,reproducingthe
maintheorem,Theorem2.5,of[17]whichrelatestotherealandcomplexcases,
andextendingittotherealquaternioncase.
Proposition 1. Let Mβ denote the set of n×N matrices with entries in F . For
n×N β
M ∈ Mβ introducethepolardecompositionaccordingto
n×N
M =U W1/2,
1
whereU ∈ Vβ andW = M†M ∈ Pβ ,Pβ denotingthesetofHermitianmatrices
1 N,n +,N +,N
with entries in F and having all eigenvalues non-negative. For g : Mβ → R
β n×N
absolutelyintegrablewehave
(cid:90) (cid:90) (cid:16) (cid:17) (cid:90)
g(M)dM =2−N U†dU (dW) (detW)β(n−N+1)/2−1
Mβ Vβ 1 1 Pβ
n×N N,n +,N
×g(U W1/2). (11)
1
Remark2. In[17]thefactor2−N ontheRHSisreplacedby2−N2 inthecomplexcase
andby2−N(N+1)/2 intherealcase,duetoadifferentnormalisationof (cid:0)U†dU (cid:1).
1 1
Thenormalisationof(cid:0)U†dU (cid:1)inherentin(11)canreadilybedeterminedbyusing
1 1
Gram-Schmidtorthogonalisationtowrite
M =U˜ T, (12)
1
whereU˜ ∈ Vβ and T is an N×N uppertriangularmatrix withnon-negative
1 N,n
diagonal entries {t }N and strictly upper triangular entries in F (in the case
ii i=1 β
β =4, T isrepresentedasa2N×2N complexmatrixwithdiagonalentries t I ).
ii 2
MATRIXPOLARDECOMPOSITIONANDTHEBLASCHKE-PETKANTSCHINFORMULA 5
Astandardchangeofvariablesformulainrandommatrixtheorytellsus(seee.g.
[18]fortherealcase,[7]or[3,Lemma2.7]forallthreecases)
N (cid:16) (cid:17)
(dM) = ∏tβ(n−j+1)−1(dT) U˜†dU˜ . (13)
jj 1 1
j=1
Multiplying both sides by e−TrM†M = e−TrT†T, taking Tr in the real quaternion
casetobe 1 ofitsusualmeaninginkeepingwith(10),andintegratinggives
2
(cid:90) (cid:16)U†dU (cid:17) =2N∏N πβ(n−i+1)/2 = ∏N σ , (14)
U1∈VNβ,n 1 1 i=1 Γ(β(n−i+1)/2) i=1 β(n−i+1)
whereσ =2πl/2/Γ(l/2)isthesurfaceareaofthe(l−1)-sphereinRl. Inthereal
l
casetheabovecalculationcanbefoundin[18,Ch.2].
Wecanmakeuseof(14)tosimplify(11)inthecasethat g(M) isindependent
ofU . Thisextendsmatrixintegrationformulasfrom[17,Th.2.5],applicablefor
1
realorcomplexentries,totherealquaternioncase.
Corollary3. For f : Pβ →Rabsolutelyintegrablewehave
+,N
(cid:90)
f(M†M)(dM)
Mβ
n×N
= ∏N σβ(n−i+1) (cid:90) f(M†M) (cid:16)detM†M(cid:17)β(n−N)/2(dM). (15)
i=1 σβ(N−i+1) MβN×N
3. Blaschke-PetkantschinFormula
3.1. Blaschke-PetkantschinFormula—MilesVersion. TheGrassmannian Gβ
N,n
isthesetofall N-dimensionalsubspacesinFn. Thissetcanbeidentifiedwiththe
β
quotientspace Vβ /O(N). Thus,sincetheelementsof Vβ are N orthonormal
N,n N,n
vectorsinFn,elementsinthequotientconsistofall N orthonormalvectorsspan-
β
ningthesame N-dimensionalsubspace. Itfollowsthatwehavethedecomposition
ofmeasure(see[10]fortherealcase)
(cid:16) (cid:17) (cid:16) (cid:17)
U˜†dU˜ =dωβ U†dU , (16)
1 1 N,n
where U˜ ∈ Vβ , U ∈ Vβ and dωβ denotes the invariant measure on Gβ .
1 N,n N,N N,n N,n
Asamatrixdecomposition
U˜ = AU (17)
1
forsome Asuchthatthecolumnsof Aforma"reference"setoforthogonalvectors
spanningtheparticular N-dimensionalsubspaceofFn. Forfutureapplicationwe
β
notefrom(16)and(14)that
volVβ N σ
volGβ = N,n = ∏ β(n−i+1). (18)
N,n volVNβ,N i=1 σβ(N−i+1)
6 PETERJ.FORRESTER
Makinguseof(16)and(17),itwasobservedbyMoghadasi[17]thatinthereal
case(11)canbeusedtodeduceanintegrationformulaequivalenttotheBlaschke-
Petkantschindecompositionofmeasureformulafromintegralgeometry[2],[20],
[1],informduetoMiles[16],. Howevertherequiredworkingisnotrestrictedto
therealcase,allowingforageneralisationoftheBlaschke-Petkantschinformula
tothecomplexandrealquaternioncases.
Proposition 4. Let g : Mβ → R be integrable. Make use of the notations Gβ ,
N,n N,n
β
dω fortheGrassmannianandcorrespondinginvariantmeasureasintroducedabove.
N,n
Wehave
(cid:90)
g(M)(dM)
M∈Mβ
N,n
= (cid:90) dωβ (cid:90) (dM)g(AM) (cid:16)detM†M(cid:17)β(n−N)/2. (19)
A∈Gβ N,n M∈Mβ
N,n N,N
Proof. Substituting(16)and(17)intheRHSof(11)thisreads
(cid:90) (cid:90) (cid:90) (cid:16) (cid:17)
g(M)(dM) =2−N dωβ U†dU
Mβ A∈Gβ N,n U∈Vβ
n×N N,n N,N
(cid:90)
× (dW)g(AUW1/2) (detW)β(n−N+1)/2−1. (20)
W∈Pβ
+,N
Ontheotherhand,inthecase n = N,andwith g(M) replacedby
(cid:16) (cid:17)β(n−N)/2
g(AM) detM†M ,
thesesamesubstitutionsin(11)tellusthat
(cid:90) (cid:16) (cid:17)β(n−N)/2
g(AM) detM†M (dM)
Mβ
N×N
(cid:90) (cid:16) (cid:17)(cid:90)
=2−N U†dU (dW)g(AUW1/2) (detW)β(n−N+1)/2−1. (21)
U∈Vβ W∈Pβ
N,N +,N
Substituting(21)in(20)gives(19). (cid:3)
3.2. Blaschke-Petkantschin Formula — Affine Version. One viewpoint on (19)
isasthedecompositionofmeasure
∏N dvn = (cid:12)(cid:12)det[vN]N (cid:12)(cid:12)β(n−N)∏N dvNdωβ , (22)
k (cid:12) k k=1(cid:12) k N,n
k=1 k=1
where vn ∈ (F )n, while vN ∈ (F )N is the co-ordinate for vn in a particular
k β k β k
basis for Span{vn}N , and dωβ is the invariant measure on subspaces (the
k k=1 N,n
Grassmannian Gβ )formedbythespan. Following[16],consider vn ∈ (F )n for
N,n k β
MATRIXPOLARDECOMPOSITIONANDTHEBLASCHKE-PETKANTSCHINFORMULA 7
k =0,1,...,N,anddefine zn = vn−vn (k =1,...,N). Thenwehave
k k 0
N N
∏dvn = dvn∏dzn
k 0 k
k=0 k=1
=dvn(cid:12)(cid:12)det[zN]N (cid:12)(cid:12)β(n−N)∏N dzNdωβ , (23)
0(cid:12) k k=1(cid:12) k N,n
k=1
wherethesecondequalityfollowsuponapplying(22)to{zn}N . WithBann×N
k k=1
matrixwithcolumnsforminganorthogonalsetwhichisabasisforSpan{vN−
k
vN}N ,wehavezn = BzN. NowdefinezN = vN−vN. Then(23)canberewritten
0 k=1 k k k k 0
∏N dvn =dvn(cid:12)(cid:12)det[vN−vN]N (cid:12)(cid:12)β(n−N)∏N dvNdωβ . (24)
k 0(cid:12) k 0 k=1(cid:12) k N,n
k=0 k=1
Finally,write vn = BvN+r,where r isanelementoftheorthogonalcomplement
0 0
⊥,β
of the column space of B, and let dS denote the volume element associated
n−N
withthissubspace. Thisreduces(24)totheaffineformof(22).
Proposition5. Wehave
∏N dvn = (cid:12)(cid:12)det[vN−vN]N (cid:12)(cid:12)β(n−N)∏N dvNdωβ dS⊥,β . (25)
k (cid:12) k 0 k=1(cid:12) k N,n n−N
k=0 k=0
Theformula(25)isthedecompositionofmeasuregivenintheoriginalworks
[2]and[20]. WithBandrasspecifiedintheabovetext,wenotethatvn = BvN+r,
k k
andinparticular
(cid:107)vn(cid:107) = (cid:107)vN(cid:107)2+(cid:107)r(cid:107)2. (26)
k k
Also, the fact that B†B = I allows us to read off from (25) an analogue of the
N
matrixintegrationformula(15).
Corollary6. For f : Pβ →Rabsolutelyintegrable,andletV = [vn−vn]. Wehave
+,N n k 0
(cid:90) N
f(V†V ) ∏dvn
vnk∈(Fβ)n n n k=0 k
N σ (cid:90) N
= ∏ β(n−i+1) f(V†V )|detV |β(n−N)dvn∏dvN. (27)
i=1 σβ(N−i+1) v0n∈(Fβ)n,vkN∈(Fβ)N N N N 0k=1 k
4. Applications
4.1. Induced Random Matrix Ensembles. Suppose the random matrix M ∈
Mβ hasPDFoftheform P(M†M). Thecorrespondinginducedensembleisde-
n×M
finedasthesetofrandommatricesoftheformK =U(M†M)1/2,whereU ∈ Vβ .
N,N
It is known [6] that the PDF of K is proportional to (detK†K)β(n−N)/2P(K†K).
Hereitwillbeshownthatthisresult,togetherwiththeproportionalityconstant,
canbededucedasaconsequenceofCorollary3.
8 PETERJ.FORRESTER
Corollary 7. Let M,U and K be specifiedas in theabove paragraph. The PDFof K is
equalto
N σ
∏ β(n−i+1) (detK†K)β(n−N)/2P(K†K). (28)
σ
i=1 β(N−i+1)
Proof. FromthedefinitionsthePDFof K isequalto
(cid:90) (cid:90)
[U†dU] (dM)δ(K−U(M†M)1/2)P(M†M), (29)
Vβ Mβ
N,N n×N
where [U†dU] denotesthenormalisedinvariantmeasureon Vβ . Makinguseof
N,N
(15)with f(M†M) = δ(K−U(M†M)1/2)P(M†M) thisisequalto
N σ (cid:90) (cid:90)
∏ β(n−i+1) [U†dU] (dM)δ(K−U(M†M)1/2)
i=1 σβ(N−i+1) VNβ,N Mnβ×N
(cid:16) (cid:17)β(n−N)/2
× detM†M P(M†M). (30)
FromthematrixpolardecompositionweknowthatforauniqueU ∈ Vβ and
N,N
with M ∈ Mβ , we have M = U(M†M)1/2. Hence in (30) (but not in (29) we
N×N
canreplacethedeltafunctionbyδ(K−M). TheintegrationoverUthusdecouples,
givingunity,whiletheintegrationover M gives(20). (cid:3)
4.2. Moments of Determinants. In the real case, a well established application
of the Blascke-Petkantschin formula (19) is to the evaluation of the moments of
detM†M in the case that the PDF for the distribution of M is a function of the
lengths(cid:107)m (cid:107),wherem denotesthek-thcolumnof M(see[16],[15,§3.1.3]). Since
k k
A in (19) has the property that A†A = I , and since the k-th column of AM is
N
Am ,weseethatchoosing g(M)tobeafunctionofthelengthsofthecolumnswe
k
have p(M) = p(AM). Makingusetooof(18)showsthatinthiscircumstancethe
Blaschke–Petkantschinformulareducesto
(cid:90)
g(M)(dM)
Mβ
n×N
= ∏N σβ(n−i+1) (cid:90) g(M) (cid:16)detM†M(cid:17)β(n−N)/2(dM) (31)
i=1 σβ(N−i+1) MβN×N
(cf.(15)).
Asimpleexampledependingonlyonthelengthsofthecolumnsiswhen
N
g(M) = ∏p((cid:107)m (cid:107)). (32)
i
i=1
Letusnormalise g(M) for M ∈ MN×N byrequiringthat
(cid:90) (cid:90) ∞
p((cid:107)m(cid:107))dm = σ rβN−1p(r)dr =1,
βN
(Fβ)N 0
MATRIXPOLARDECOMPOSITIONANDTHEBLASCHKE-PETKANTSCHINFORMULA 9
wherethefirstequalityfollowsbyconvertingtopolarcoordinates. Soifwechoose
p(r) tobeuniformfor0<r <1andzeroforr >1,tobecorrectlynormalisedwe
shouldset
βN
p(r) = χ0<r<1, (33)
σ
βN
where χ = 1 for S true and χ = 0 otherwise. And requiring each m to have
S S i
lengthunityisachievedbychoosing
1
p(r) = δ(1−(cid:107)m(cid:107)). (34)
σ
βN
AlsoofinterestistheGaussiandistribution
P(M) = (cid:16)1(cid:17)βN2/2e−TrM†M := (cid:16)1(cid:17)βN2/2e−∑lN=1(cid:107)ml(cid:107)2 (35)
π π
(thisintherealquaternioncaseTrisdefinedas 1 itsusualvalue;cf.(10))forwhich
2
p(r) = (cid:16)1(cid:17)βN/2e−r2. (36)
π
ThecorrespondingmomentsofdetM†M canbereadofffrom(31).
Proposition 8. Let the PDF of the N×N matrix M be of the form (32) with p(r)
specifiedby(52). Wehave,form ∈ (F )N, (i = 1,...,N),andqsuchthatbothsides
i β
arewelldefined
(cid:16)βN(cid:17)N(cid:90) |detM|q(dM) = ∏N σβ(N−i+1) (cid:16)Nσβn(cid:17)N(cid:12)(cid:12)(cid:12) (37)
σβN (cid:107)mi(cid:107)≤1 i=1 σβ(n−i+1) nσβN (cid:12)n=N+q/β
and
(cid:16) 1 (cid:17)N(cid:90) |detM|q(dM) = ∏N σβ(N−i+1) (cid:16)σβn (cid:17)N(cid:12)(cid:12)(cid:12) . (38)
σβN (cid:107)mi(cid:107)=1 i=1 σβ(n−i+1) σβN (cid:12)n=N+q/β
IntheGaussiancase(35),
(cid:16)1(cid:17)βN2/2(cid:90) e−∑lN=1(cid:107)ml(cid:107)2|detM|q(dM)
π (F )N
β
= ∏N σβ(N−i+1) (cid:16)1(cid:17)βN(N−n)/2(cid:12)(cid:12) . (39)
(cid:12)
i=1 σβ(n−i+1) π n=N+q/β
Proof. Theonlypointthatremainsistojustifyextendingfromvaluesq = (n−N)β
withn ∈Z≥0tothegeneralvaluesofq. ThiscanbedonebyappealingtoCarlson’s
theorem(seee.g.[7,Prop,4.1.4]),whichtellsisthatiftwofunctions,analyticin
therighthalfplane,agreeattheintegersandareboundedbyO(eµ|z|), µ < π,in
(cid:3)
thisregionthenthefunctionsareidentical.
10 PETERJ.FORRESTER
Remark9. Intherealcase(35)correspondstoaresultofWilks[25];themethodof
derivationpresentedhereisthatof[17]. Foreach β =1,2and4both(38)and(39)
canbefoundin[21],wherethederivationmadeusetheJacobianforaQRtype
decomposition; see also [14, 15]. Introducing polar coordinates m = r q when
i i i
(cid:107)q (cid:107) =1in(46),thedependenceonr canbeintegratedoutand,afteruseofthe
i i
explicit formula for σ below (14), (38) reclaimed. This same procedure can be
l
usedin(37)toprovideanotherderivationof(38).
Remark 10. Differentiating both sides of (37)–(35) with respect to q and setting
q =0providesuswiththeevaluationof (cid:104)log|detM|(cid:105). IntheGaussiancase,this
averagehasappearedbefore,inthecontextofcalculatingtheLyapunovexponents
fortherandommatrixproduct
L = M(t)M(t−1)···M(1), (40)
t
whereeach M(j) ischosenfromthedistributionof M [19,8,9]. Inrelationtothis
problem,onerecallsfromthemultiplicativeergodictheoremofOseledecthatthe
limitingmatrixVN :=limt→∞(L†tLt)1/(2t) iswelldefined. TheeigenvaluesofVN
areallnon-negative,andwhenwrittenintheform {eµl}lN=1,theexponents µl are
theLyapunovexponents. Let u beaunitvector. Inthecasethateachcolumnof
M(j) in(40)dependsonlyonitslength,thedistributionof u†M(j) isindependent
of u, and a result of Raghunathan relating to the partial sums of the Lyapunov
exponentstellsusthatinthissetting
N
∑µ = (cid:104)log|detM|(cid:105). (41)
i
i=1
TheresultsofProposition8whenusedintheRHSof(15)provideuswithan
extensionofthemomentformulastonowapplytodetM†M when M = [mn]N
k k=1
isan n×N matrixwithentrieswithentriesinF . Intherealcase,thiswasfirst
β
noticedandmadeexplicitin[16].
Corollary11. With Mspecifiedasabove,let M = [mN]N correspondtothe N×N
N k k=1
case. Letthenotation(cid:104)·(cid:105) refertotheexpectationwithrespecttothePDFrelatingto#.
#
Set A = ∏N σ /σ . Wehave
β,n,N i=1 β(n−i+1) β(N−i+1)
(cid:68)(detM†M)h/2(cid:69) = A (cid:16)nσβN(cid:17)N(cid:68)|detM |h+β(n−N)(cid:69)
(cid:107)mnk(cid:107)≤1 β,n,N Nσβn N (cid:107)mkN(cid:107)≤1
(cid:68)(detM†M)h/2(cid:69) = A (cid:16)σβN(cid:17)N(cid:68)|detM |h+β(n−N)(cid:69)
β,n,N N
(cid:107)mnk(cid:107)=1 σβn (cid:107)mkN(cid:107)=1
(cid:68) (cid:69) (cid:16)1(cid:17)βN(n−N)/2(cid:68) (cid:69)
(detM†M)h/2 = A |detM |h+β(n−N) ,
β,n,N N
Gn π GN
whereinthefinalequationthesubscriptsGn andGN refertotheGaussiandistribution
on Mand M .
N