This research was supported by the National Science Foundation under
Grant No. GU-2059, the U.S. Air Force Office of Scientific Research under
Contract No. AFOSR-68-l4l5 and the Sakko-kai Foundation.
ON THE DISTRIBUTION OF THE LATENT ROOTS OF A COMPLEX
WISHART MATRIX (NON-CENTRAL CASE)
by
TAKESI HAYAKAWA
University of North Carolina and
Institute of Statistical Mathematics
Institute of Statistics Mimeo Series No. 667
University of North Carolina at Chapel Hill
FEBRUARY 1970
Distribution of Complex Wishart Matrix
Abstract
This paper considers the derivation of the probability density function
of the latent roots of a non-central complex Wishart matrix. To treat this
problem, we define the generalized Hermite polynomials of a complex matrix
argument and give some properties of the generalized Hermite polynomials.
By using the generating function of the generalized Hermite polynomials, we
can obtain the exact probability density functions of the latent roots,.
the maximum latent root and of trace of the latent roots of a non-centr~l
complex Wishart matrix.
Classification number:
10.
Main words and notations
H(T):
K
r::y(S):
K
generalized Hermite polynomial
argument T.
generalized Laguerre polynomial
trix S.
(g.H.p.)
(g.L.p.)
of a complex matrix
of a hermitian ma-
d(U):
The unitary invariant measure with total volume unity.
d*(U):
The unitary invariant measure.
d(L) :
The semi-unitary invariant measure with total volume ~nity.
C (A):
The zonal polynomial which corresponds the partition
into not more than m parts.
etr A:
exp(tr A).
det A:
determinant of
K
A.
K
of
k
On the distribution of the latent roots of a complex
Wishart matrix (non-central case)
by
Takesi Hayakawa
University of North Carolina and
Institute of Statistical Mathematics
I.
INTRODUCTION
N. G. Goodman [3], C. G. Khatri [7], [8] and A. T. James [6] have dis-
cussed some distribution problems of the complex multivariate normal distribution which appears in time series analysis and physics.
The problems
of complex variates can be treated in the same way as those of the real
variates in the case of normal distributions.
In this paper, we consider
the distribution problems of the latent roots of a positive definite random
Hermitian matrix which are extensions of author [5] to the complex variates.
We introduce the generalized Hermite and Laguerre polynomials with complex
matrix argument to handle these problems.
2.
NOTATIONS AND USEFUL RESULTS
We shall use the following notations which are given by James [6].
(1)
(2)
r (a)
m
=
r (a,K)
m
~~m(m-1)
=
~
~m(m-1)
m
f(a-a+1).
II
a=l
m
f(a+k -a+1).
a
IT
a=l
m
(3)
[a]
K =
IT
a=l
(a-a+1)k
a
= r (a,K)/r (a).
m
m
This research was supported by the National Science Foundation under
Grant No. GU-2059, the U.S. Air Force Office of Scientific Research under
Contract No. AFOSR-68-l4l5 and the Sakko-kai Foundation.
",
2
where
K = (kl, .•• ,k )
m
is a partition of
k
into not more than
m parts.
The corresponding generalized hypergeometric functions are defined as
'"F (m)
(a , .•• ,a ,b , ••• ,b q ,A,B)
p l
'p q
l
(4)
00
L L
=
C (A)
...
K [b l ]K
k=O
where
[al]K
[ap]K
C (A)C (B)
K
K
[b q]K
klC (I )
K m
is a zonal polynomial of a Hermitian matrix
K
to a partition
K of
'" (m)
F
( .•.. , .... ; A, I)
q
P
k.
If either
= P'"Fq (...,
A or
B is
I,
A corresponding
then we write as
m
...; A).
The most fundamental properties of a zonal polynomial are of the same
form as in the real case, so that
(5)
C (A)C (B)
fU(m)
CK (AUBU') d(U)
f
etr(XU+U'X')d(U)
K
=
K
C (I )
K
m
and
( 6)
=
oFI(n;xX'),
U(n)
where
d(U)
in (5) and (6) is the invariant measure normalized to make
the't:Cl,'t:C3.,lmeasure unity on the unitary group
~\~';':
an~:~,'
respectively,a.nd
an arbitrary
nxn
Z mxn (m6n),
mn
_",TI~_
r
(n)
m
U(n)
of order
m
X is
complex matrix.
be a probability density function (p.d.f.)
then the p.d.f. of the Hermitian matrix
by
(7)
and
Aand/B' are Hermitian matrices, and
Hsu's Lemma. Let f(ZZ')
of
U(m)
(detR)n-m f(R).
R=ZZ'
is given
3
The p.d.f. of the latent roots
of
R is also given
by
m(n+m)
n
( 8)
f
(detA)n-m f(A)
(n)f (m)
m
m
2
(A.-A.) ,
1.
J
i<j
IT
= f(A).
if
f(UAU')
3.
GENERALIZED HERMITE POLYNcr-1IALS
Hayakawa [5] defined the generalized Hermite polynomial (g.H.p.) with
a real matrix argument in order to discuss the distribution problems of the
latent roots of the non-central Wishart matrix.
In this section,
we show that they can be extended to the complex variate case and they can
be treated analogously as the real variate case,
,..,
Here we define the g.H.p.
nomial (g.L.p.)
k,
L (S),
Y
K
>
H (T)
K
-1,
and the generalized Laguerre poly-
which correspond to the partition
in the following way:
etr(-TT')H (T)
(9)
K
=
(_l)k
mn
n
where
T
and
Ware
mxn
J etr(-i(TW'+wT'»etr(-wW')C (wW')dW,
K
W
(m~n)
arbitrary complex matrices, and for
Y > -1,
(10)
etr(-S)LY(S)
K
=
J
R'=R>O
AY(RS)(detR)Yetr(-R)CK (R)dR,
K
of
4
where
Sand
Rare
mxm
~
Hermitian matrices and
~
~
r (y+m)A (RS)
m
::::
y
OF 1 (y+m, -RS) ,
[2] •
The next theorem gives a relation between g.H.p. and g.L.p.
Theorem 1.
H(T)
(11)
(_l)k
K
Proof.
LU-m (TT').
K
From the definition of g.H.p. and the Hsu's Lemma in the complex
case, we have
etr(-TT')H (T)
::::
K
(_l)k
mn
fWfU(n)
7T
::::
(_l)k
mn
fW
7T
::::
etr{-i(wuT'+TU'W')}etr(-wW')C (wW')dWd(U)
K
~
OF1 (n; -TT'wW')etr(-wW')C K (wW')dW
JR'::::R>O An-m (TT'R) (detR)n-m etr(-R)C (R)dR
(_l)k
K
::::
- :-'Il-m (IT' )
(_l)k etr( -TT')L
K
,
which completes the proof.
We can show easily the following corollaries from the definition (8)
and Hsu's Lemma.
Co ro 11 ary 1.
ifK (T)
(12)
where
U
l
€
U(m)
::::
and
U
2
€
U(n),
respectively.
5
Coro 11 ary 2.
(13)
IiiK
(T) I :,; K
etr(TT')K
[n] C (I),
(14)
iiK (0)
Theorem 2.
(Generating function of g.H.p.' S )
Let
(-l)k[n]
=
Sand T be
f f
(15)
K
etr(TT')C (I),
K
mxn (m:';n)
arbitrary complex matrices, then
etr{-SS'+UlSUZT'+TUiS'Ui} d(Ul)d(U Z)
U(m) U(n)
00
L L
k=O
where
U
l
E
U(m)
and
ii
(T)C (SS')
K
K
C (I)
k! [n]
KKK
U
zE
U(n),
respectively.
The right hand side of
(15) converges absolutely.
Proof.
We show (15) by the direct method, using (5), (6) and (9).
00
R.H.S.
= etr(TT')
:n
7T
f
\.]
(_I)kC (SS')
I L kl [n]
K
~
etr{-i(TW'+wT')}·
k=O K K K (I)
C
• etr(-wW')C (wW')dW
K
00
=
etr(TT' )
mn
7T
fW fU(m)
(_l)k
L L k! [n] K CK (SS'U1WW'U')·
1
k=O
K
• etr{-i(TW'+wT')}etr(-wW')dW
=
etr(TT')
mn
7T
fW fU(m) fU(n)
etr{i(W'U SU +U'S'U'W)} •
1 Z Z 1
• etr{-i(TW'+wT')}etr(-wW')dW
6
=
fU(m) fU(n)
The absolute convergence of (15) is shown by using (13).
This completes
the proof.
Theorem 3.
Let
(Generating function of g.L.p.' S )
Sand
Z be
mxm
f
det(1-Z)-y-m
(16)
positive definite Hermitian matrices, then
etr(-SUZ(1-Z)-lU')d(U)
U(m)
00
LY(S)C (Z)
L L
k=O
liz/ I
where
roots of
Proof.
K
K
K
II Z II
k!C (I )
K
<
1,
m
means the maximum of the absolute values of the characteristic
Z.
Similar to Theorem 2, using (5), (7) and (10).
~
Remark 1: The invariance of HK (T) with respect to U(m)
from left and
U(n)
and
from right is obvious from Theorem 2, because
invariant unitary measures on
Remark 2:
n
ron
(detZ)
-n
Let
y = n-m
-
etr(-SS'Z)
(16) by replacing
Theorem 4.
and
U(n),
d(U )
2
are
respectively.
The relation between the generating functions of the g.H.p. 's
and the g.L.p. 's when
by
U(m)
d(U )
l
Z with
is as follows.
Multiply both sides of (15)
and integrate with respect to
S.
Then we have
-Z.
(Mehler's formula)
Sand
T be an
mxn
arbitrary complex matrix.
Then
7
2
fU(n)
(17)
00
L
~
2k
uk!
k --O
Proof.
etr{- __
u__ (SS'+TT') +
u_ (U SU T'
2
l 2
l-u
l_u 2
~
H (T)H (S)
L
K
lui
K
< 1.
[n] C (I m)
KKK
The proof is exactly similar to that of Theorem 2.
Corollary 4.
L r;n-m
(18)
L~-l (tr TT').
(TT')
K
K
Proof.
If we set
S=O
in (17), we have
00
____71
(19)
(1_u 2 )mn
since
2
e t r [- __
u __ TT' ]
2k
u
l_u2
k=O
k!
(_l)k
L H (T),
K
K
HK
(0) = (_l)k[n] C (I).
KK
The left hand side of (19) is a generating function of a univariate
Laguerre polynomial
L~n-l(trTT'), i.e.,
00
L.H.S.
2k
u
k!
Lmn - l (tr TT').
k
k=O
Hence by comparing the coefficients of
u
2k
and by Theorem 1, we have (18),
which completes the proof.
We can obtain a more general formula for the g.L.p. 's than (18).
8
Co ro 11 ary 5.
= Lm(y+m)-1 (tr S),
L LY(S)
K
(20)
k
K
where
Proof.
S
is an
mxm
positive definite Hermitian matrix.
From the definition (10), we have for
Ixl
<
1
00
etr(-S)
\'
L
k
~
k!
k=O
\' LYK (S)
L
=
f_
R'=R>O
K
AY(RS) (detR)Yetr (-(I-x)R)dR
00
1
r m(y+m)
L
1
k!
L
K
k=O
. f
1
[y+m]
K
etr(-(I-x)R) (detR)YCK(-RS)dR
R'=R>O
00
1
=
(I-x)
(y+m)m
L
k=O
-l \' C (2)
k! L K I-x
K
1
etr(- __1__ S).
(l_x)m(y+m)
I-x
Hence
00
(21)
\'
L
k=O
k
~
k!
\' LYK (S)
L
etr(- ~ S)
1
I-x
(l_x)m(Y+m)
K
00
k
~ L m(y+m)-1 (trS),
L k! k
k=O
\'
By comparing the two sides of (21), we obtain (20).
Ixl
<
1-
9
4.
JACOBIANS OF THE LATENT ROOTS AND INTEGRALS
In this section, we give some useful transformations for finding the
p.d.f. of the maximum latent root of a positive definite Hermitian matrix,
and of the latent roots of a non-central complex Wishart matrix with known
covariance.
Lemma 1.
We also give some related Beta-type integrals.
Let
We decompose
where
U
variables
cept
ull~
S
S
is an
be an
mxm
positive definite random Hermitian matrix.
as follows,
mxm
R
unitary matrix which has only
R
I
I
independent
the first column elements of
u 2l '···,uml ' u 2l '···,uml '
and V isan (m-l) x (m-l)
2 (m-l)
U
ex-
positive definite random Hermitian
matrix which ranges
Then
m
(23)
2
det(AII-V) dA1dV
dS
IT
R
I
dualdual
a=2
Proof.
Note:
The proof of this lemma is given in Appendix I.
If we decompose
(24)
where
S
U
v
S
further such that
=
is a unitary matrix of order
contains independent variables, and
trix which ranges
AkI
>
Vi<.
=
V
k
>
V
k
(m-v+l)
is an
0, --then
whose first column vector
(m-k) x (m-k)
Hermitian ma-
10
k
dS
(25)
IT
=
i<j
(A.-A,)
1.
J
2
This can be shown, by induction, from Lemma 1.
Lemma 2:
Let
X be an
mxn
(m~n)
arbitrary complex matrix,
unitary matrix whose diagonal elements are all real values, and
nxm
(m~n)
semi-unitary matrix (i.e.,
all complex variables, and
m
xX'.
where the
is given by
dX
(27)
where
Proof.
Note:
d(L)
~TI
m(n+tn-l)
~
(detli)2(n-m)+1
r
(n)r (m)
m
m
IT
i<j
is a semi-unitary invariant measure.
The proof is shown in Appendix II.
If we decompose
X
then we have
an
are all
Then the Jacobian of the decomposition
Uli1'
X
L
mxm
= I m) whose diagonal elements are
Ii = diag (A 1 , ••• ,A ),
real-valued latent roots of
(26)
1'L
U an
X as follows:
-, ,
L
11
where
=
c
'IT
mn
r m (n)
Theorem 5.
(28)
f
_
1
m-l
(detW)a-mdet (1-W)2 CK(l W)dW
>W'=W>O
~
~
r (a)r (m)r(m)
m
m
'IT
(29)
m-l~
r (a+m)
m
m
f
m
( TI w.) a-m TI (l-w.) 2
i=2 1
i=2
1
TI
(w. -w.)
2<.<.<
1
J
_1 J_m
2
C
1
K
W2
.
•
[
]
w
m
m
TI dw.
1
i=2
~
~
r (a)r (m)
=
m
m
2
[a] K
~
(ma+k) -=-[a-+-m-::"]- CK(1 m).
K
Proof.
The following formula is well-known:
~
(30)
~
r (a)r (m)
f
(detS)a-m CK(S)dS
m
r (a+m)
1>8'=S>0
To prove (28), we decompose
m
m
S
as follows:
Then from Lemma 1, the integral element can be written as
•
12
dS
Hence inserting these results in (30) , and noting that
m
fm
IT
R2 in 1 2
.L uil+.L uil:,>l
~=2
i=2
R
1
duildu il
7f
1
m-l
and
f(m)
f
Al ma+k-l dAl
1
ma+k
0
~=2
we have (28).
W
To prove (29), we decompose
U
2
E
W further, so that
U(m-l),
Then
where
d*(U )
2
over
U(m-l),
is an invariant measure on
U(m-l).
Hence by integrating
we have (29), since
7f
(m-l) (m-2)
~
f m-l (m-l)
5.
THE DISTRIBUTION OF THE LATENT ROOTS OF A CO~1PLEX NON-CENTRAL WISHART
MATRIX WIll-l KNOfIN COVARIANCE
The probability density function of the latent roots of a complex noncentral Wishart matrix was given by James [6].
However, in some cases it
is not convenient to treat the distribution problems of the related statistics.
We here give another formula, expressed in terms of g.H.p.'s.
13
Theorem 6.
Let
X mxn (m:::;n)
be distributed with p.d.f.,
(31)
then the p.d.f. of the latent roots of
m(m-1)
_~_1T_ _~_ _ etr(-E -l MM ,) (detA)n-m
(32)
r (n)r (m)
m
o
det(xX'-AE)
m
is given by
IT (A.-A.) 2 •
1.
J
i<j
~
00
L L
k=O
where
•
Proof.
0:
-h:
~
M) C (A)
K
~ K
k! [n] C (I )
2
m
KKK
A = diag(A1, ••• ,A ).
m
E-tX in (31) as follows:
We decompose
(33)
where
H
h:-
U A2 L'·
1
Y
U , L
1
and
Then
m(m+n-1)
2
~
(detA)n-m IT (A.-A.) dAd(U )d(L).
1
r (n)r (m)
i<j
1.
J
~1T
dY
(34)
A are same matrices as those of Lemma 2.
m
m
Hence by inserting (33) and (34) into (31), we have the joint p.d.f. of
A, U
1
(35)
and
L;
m(m-1)
_~~1T_ _~_ _
etr( -E
-1
MM') (detA)
n-m
r (n)r (m)
m
m
h: -
IT <A.-A.) 2 •
1.
J
i<j
h:
h: -
-
• etr(-A 2 L'LA 2 +U A2 L'M'E
1
then
L'U (L'U )' = I
22m
-h:
2
+E
-h:
2
h: -
MLA 2 Ui) .
and the semi
14
unitary invariant measure
del)
is unchanged by the unitary transformation.
VI
Therefore, we integrate (33) with respect to
f_
*:2-
m
*:2- _1--*:2
*:2MLA V')d(V )d(L)
I I I
V(m)
JL'L=1 f
m
V(m)
*:2-
f
*:2-
-
etr(-A L'LA +V A L'V M'E
1
-*:2
2
and
V(m)
form as the generating function of g.H.p. 's if we set
= E-~M.
+E
-*:2 -
*:2-
MV'LA V') •
2
1
V(n)
The integral of the R.H.S. with respect to
T
L.
etr(-A L'LA +V A L'M'E ~+E
f
L'L=1
*:2
and
V(n)
is the same
S = A~L'
and
Thus we have
~
00
H (E
-*:
J1'L=1 m k=OI I k! [n]
R.H.S.
2
~
M)C (A)
K
del)
K
C (I )
KKK
m
00
~
H (E
K
-*:
2
~
M) C (A)
K
which completes the proof.
Corollary 6. If we set M=O in (32), then we have the p.d.f. of A in the
central case from (14) [1].
Theorem 7.
Let
A be distributed with the p.d.f. (32).
maximum latent root
Al
of
A is given by
Then the p.d.f. of the
15
00
~
r
(36)
r
Proof.
(n)
m
etr(_E-IMM')A~n-1 L (mn+k) Aki \'
k!
L
(n+m)
K
m
k=O
If we set
A. = Aiw.
1
1
W2,""w ,
m
respect to
(i = 2, .•• ,m)
[n+m]
K
in (32), and integrate it with
then from (29) we have (36) immediately.
Corollary 7. The c.d.f. of Al is given by
00
~
Pr{Al<X}
(37)
=
rm(n)
~
r m(n+m)
etr(-E
-1 -
mn
L
MM')x
x
k
k!
k=O
~
H
L
K
0:
-k
2 M)
[n+m]
K
K
Theorem 8.
Let
A be distributed with p.d.f. (32).
Then the p.d.f. of
T = trA
is given by
(38)
1
r(mn)
etr( "';L;-lMM' )T mn - 1
K
Proof. We can obtain (38) as in the proof of Theorem 5 in [5].
Remark:
where
(38) is of the same form as the p.d.f. of the
is a non-central chi-square variable of
freedom with non-central parameter
0 2 = trE-IMM'.
T = 2X~mn(02),
2mn
degrees of
Since from Theorem 1
and Corollary 4, the series term can be represented by
k! (mn)k
k=O
this can also be represented using a Bessel function as
16
However, as the Bessel function
Jmn_l(2(-T02)~)
is written as
1
f(mn)
the p.d.f. of
T
is written as
00
1
2
mn-l
f(mn) exp(-o -T)T
L
k=O
This is the p.d.f. of
(To 2)k
k!(mn)k
17
APPENDI X 1.
(PROOF OF
LEMvlA 1,)
Differentiating both sides of (22), we have
,
(Al.l)
dS
Multiply by
U'
U from the right to obtain
from the left and
Al
UrdU
U'dSU
since
that
UrdU = -dU'U.
dW
ates and
dAl
J
By setting
+
II
J JU'dU'
dW = U'dSU
and
dT = UrdU,
we can see
is a Hermitian matrix whose diagonal elements are all real varidT
is a skew Hermitian matrix whose diagonal elements are all
pure imaginary.
(AI. 2)
dW
(AI. 3)
dT
Hence we have
=
UrdU.
To obtain the Jacobian of this transformation, we shall use the exterior
product of the differentials.
First of all, we know that
m
R
I
rr a= IdS a~orr a>~odS a~odS a~a' [7].
the exterior product of
matrix
U has only
as follows
dT,
2(m-l)
Before we calculate
we must consider the structure of
independent elements.
U.
The
Hence we represent
U
18
run
(AI. 4)
U =
R
I
~l+ial
l
1
R I
ul+i.!!l
U~Z+iU~Z
1
m-l
m-l
where
.Q. I
,
(AI. 5)
I m- l'
I
and
are vectors of independent variables, and
.!!l
are functional vectors and matrices of
R
~l'
I
.!!~ and .!!l'
respectively.
Let
dt
dT
=
ll
1
· dT Z
·
d~~+id~~ ·
1
=
m-l
m-l
1
dUll
.
U'
dU
Z
R . d I:
d~l+J. .!!l:
1
m-l
m-l
then
Since
that
dT
dt
is a skew Hermitian matrix,
I
We will show later
can be represented by the linear combination of
ll
R
R
dtZl,···,dt ml ,
R
I
Therefore, we need only dtZl,""",dt ml " As the first colZ
R' R
I' I
umn has a constraint such that ull + .!!l .!!l +.!!l.!!l = 1, the differential
I
I
dt Zl '··· ,dt ml ·
19
of the first column of
dU
is represented as follows;
R
d~l
0d I
~l
+
1
Hence we have
d.!.l
-
R
d~l
=
(AI. 7)
I' R
1
-- a u '
u
-1-1
I
d.!.l
I' I'
R
-1- a u
U '
u -1-1 + 22
11
I'
U
22
11
Thus by forming the exterior product of
m
(AI. 8)
II
1
R I'
U
- -- a ' u
u -1-1 + 22
11
1
R R
R
- a 'u ' + U '
u - -1-1
22
11
R
dt
IT
a=2
R
a1
m
IT
a=2
dt
dt
R
,.
21
0
1 R' R'
R'
-~~1 ~1 +U 22
11
I
•
,dt
I
,
ml
1
we have'
R ' I'
al
1 I I R'
I I
~~1 ~1 -U 22
11
We can easily rewrite the determinant (say
un
1
R
~1
1
R
- det ~1
= u
11
U22
~l
U
22
I
From (AI. 5) ,
I
det
1 I' I I R '
~~1 ~1 +U 22
11
a=2
J
I
- ~~1 ~1 +U 22
11
m
IT
(AI. 9)
I
d~1
R
I
I
-a
-1
I
-U 22
R
U22
J )
1
as
R
dUal
m
IT
a=2
I
dUal.
20
0'
1
J2
(Al.lO)
1
u
1
2
I I'
I - .!!l.!!l
R I'
det 0
ll
0
1
I R'
.!!lu l
R R'
I - .!!l.!!l
ul.!!l
I'
1
=
0'
.!!l
I
- - det .!!l
2
un
R
-u
-1
R'
-u
-1
I
0'
0
I
=
l.
=
exists a unimodular matrix
(Al.H)
dt
I
ll
of
R m
m
From this calculation,
I
nex= 2 duex lIT ex= 2 duex 1·
Hence there
D such that
=
dT
is represented by
=
(-
I'
.!!l '
R'
.!!l )
[d~:]
d.!!1
=
I'
(- .!!1 ")
R'
.!!1 )
-1
D
d~:J.
d~1
Therefore
assertion.
is a function of
which proves the previous
21
Secondly, we see the relation between
Here we must also note that in
I
d~l.
In fact,
U2
dT,
dT 2
.
is a functional matr1x
dW
d~
·1 I
+
1
r=2
By setting
afk..e. ] and
•. , --RdU
m1
f~..e. + if~..e.'
we have
=
Therefore we have from (Al.ll),
ij'dU
2
dAl' dV
and
can be represented by
0
f
(1 ~
Therefore
and
d
uR
l an
k
~
m,
I
~
d~~ and
.
1.e.,
~l'
2
dT.
..e.
~
m).
22
where
I
and
dV
(D
d~l'
and
-1
)ij
(i,j
= 1,2)
-1
is a submatrix of
which establishes our assertion_
D
corresponding to
R
d~l'
Hence we need only
d/'1-
From (AI. 2) ,
R
=
R R
I I
(AII-V )d~l + V d~l '
d~l
=
I I
R R
- V d~l + (AII-V )d~l'
dW
=
dT
d~l
(AI. 12)
I
22
22
V + dV - VdT
,
22
where
dWll
dW
and
=
dT
[ d~l
Hence, forming the exterior product, we have
m
J dA l
2
1
J
2
=
=
m
R
I
IT dt
IT dt
a1
a1
a=2
a=2
det
a
a
a
-
0'
a>B~2
0'
VI
AII-v R
-VI
A1I-V R
*
det(AII-V)
*
2
-
dV
IT
0'
a
a
I
R
aB
IT
a>B~2
dV
I
aB
23
Summarizing these results, we have the Jacobian of the transformation
as
J = det(A 1 I-V)2,
which completes the proof.
II, CTHE
APPENDI X
PROOF OF LErvNA
2,)
Differentiating both sides of (26), we have
(A2.l)
Let
dX
B
nx(n-m)
trix.
=
dUAL' + UdAL' + UAdL' •
be a semi-unitary matrix such that
Multiplying both sides of (A2.l) by
U'
[L:B]
is a unitary ma-
from the left and
[L:B]
from the right, we have
=
[U'dUA:O] + [dA:O] + [AdL'L:AdL'B].
(A2.2)
U'dX[L:B]
By setting
m n-m
dF = [dF :dF ]m
2
1
dT = U'dU,
dP = -L'dL
and
dQ
is a skew Hermitian matrix and
dT
-1' dB,
have
(A2.3)
dTA + dA - Ad?
(A2.4)
- AdQ.
Here we must note that
elements.
dT
= U'dU.
dT
... ,
However,
dT
dT
has only
2
m -m
independent elements since
Therefore we can assume that the diagonal elements
are represented by another element of
R
I
dt m,m- l' dt 21 ,
... ,
I
dtm,m- 1).
has
dT,
i.e.,
Hence we have from (A2.3)
2
m
we
Z4
df(I)R
dA
CW,
a
... ,
df(I)R
0.13
R
I
dt m,m- l' dt Z1 '
=
... ,
dt
I
)
m,m-l
dp
I
0.0.
},
(a> (3),
A
a
I
(a > (3),
dt alJQ '
(a> (3),
df(I)I
So.
since
dT
(a > (3)
=
and
dP
are skew Hermitian matrices.
,
Hence, forming the ex-
terior product, we have
(AZ.5)
II
0.>13
222
R
R
I
I
II (A -A)
II dPaS dt aS II dP aS dt aS .
0.>13
a 13
0.>13
0.>13
We have from (AZ.4)
df(Z)R
as
=
R
- Aa dq a 13'
df(Z)I
as
I
- Aa dq a S.
Hence
n-m m
II df(Z)Rdf(Z)I
II
as
as
S=1 0.=1
Therefore we have
=
(
~
0.=1
Aar
Cn m n-m m
I
R
- ) IT
IT dqaS dqaS·
S=1 0.=1
25
def
dF
-
n
TI
m
R
1
TI df
df
aB
as
B==l a=l
=
r m r<n-m)+1
l
TI Aa
a=l
TI (A 2 _A 2 )2
a B
a>B
m
I
R
I
R
TI dA
TI dt
TI dPaB dPaB
dt
a
aB
aB
a=l
a>S
a>S
m
II
a=l
dPI
aa
n-m
m
I
R
TI dQaS
dQaS
B=l a=l
II
Since
TI
R
I
Qdt Qdt Q
a>1J alJ alJ
is an invariant measure on a unitary group, and
is an invariant measure of a semi-unitary matrix group, we have finally
def
(A2.6)
(detA)2(n-m)+1
dX
TI
a>S
• dAd*(U)d*(L)
•
d*(L) = TIm dp1 TI
a=l aa a> 13
J
U(m)
d*(U)
'IT
m(m-1)
r
and
m
(m)
J_
L'L=1
d*(L)
m
'IT
mn
r (n)
m
Q.E.D.
26
REFERENCES
[1]
Bronk, B. V. (1965).
Jour. Math.
"Exponential ensemble for random matrices."
6, 228-237.
Physics~
[2]
Constantine, A.G. (1965). "The distribution of Rotelling's generalized T~." Ann. Math. Statist.~
215-225.
[3]
Goodman, N.R. (1963). "Statistical analysis based on a certain
multivariate complex Gaussian distribution." (An Introduction),
Ann. Math. Statist.~
152-176.
37,
34,
[4]
Rayakawa, T. (1967). "On the distribution of the maximum latent root
of a positive definite symmetric random matrix." Ann. Inst.
Statist.~ Math.~ Tokyo~
1-17, Errata (1969).
19,
[5]
Rayakawa, T. (1969). "On the distribution of the latent roots of a
positive definite random matrix I." Ann. Inst. Statist. Math.~
Tokyo~
1-21.
21,
[6]
James, A.T. (1964). "Distributions of matrix variates and latent
roots derived from normal samples." Ann. Math. Statist.~
475-501.
35,
[7]
Khatri, C.G. (1965). "Classical statistical analysis based on a
certain multivariate complex Gaussian distribution."
Ann. Math. Statist.~
98-114.
36,
[8]
Khatri, C.G. (1966). "On the distribution problems based on positive
definite quadratic functions in normal vectors." Ann. Math.
Statist.~
468-479.
37,