709
Internat. J. Math. & Math. Scl.
VOL. 13 NO. 4 (1990) 709-716
THE DITTERT’S FUNCTION ON A SET OF NONNEGATIVE MATRICES
SUK GEUN HWANG and MUN-GU SOHN
Department of Mathematics
Teachers College
Kyungpook Unlvers Ity
Taegu 702-701
Republic of Korea
SI-JU KIM
Department of Mathematics Education
Andong Unlverslty
Andong, Kyungpook 760-380
Republic of Korea
(Received November 6, 1989 and in revised form January 12, 1990)
For X
l,....,rn),
K
Let (X)
r
all X
for
n nonnegative matrices wlth entry sum n.
Let Kn denote the set of all n
sum
vector (r
with
row
BSTRACT.
le
+
I
Dittert’s conjecture asserts that
perX.
c]
wih
K
vector (c
sum
column
equality
Iff X
[I/n]nxn.
Thls
(X)
paper
n
properties of a certain subclass of K n related to the function
conjecture.
2
l,...,cn),
n
n!/n
investigates
some
and the Dittert’s
Permanent, Dittert’s function, h-admlssable matrix.
KEY WORDS AND PHRASES.
15A48.
1980 AMS SUBJECT CLASSIFICATION CODE.
INTRODUCT ION
n nonnegatlve matrices whose entries have sum n,
Let Kn denote the set of all n
and let
. .
denote a real valued function of K n defined by
n
n
,(X)-- II
i=1 j=1
for X
[xij] e Kn
n
xlj
+
n
II
j=1 i=1
xlj-
perX
perX
where perX stands for the permanent
oES
of X;
Xno(n)"
Xlo (I)
Let J n denote the n x n matrix all of whose entries are I/n.
Is a conjecture due to Eric Dittert.
CONJECTURE (Marcus and Merris [I], Conjecture 28]).
(A) < 2
n
n
n
with equality If and only if A
Jn"
For A
For the function # there
Kn,
SUK GEUN HWANG, MUN-GU SOHN AND SI-JU KIM
710
In this paper, we will calf q the DiLLerL’s function.
For
[3]).
X e K
a matt[
n
It is proved that the
[I],
Dittert’s conjecture is true for n 6 3 (Marcus and Merris
Sinkhorn
[2], and Hwang
whose column sum
(rl,...,r)and
n
whose row sum vector is
(c l,...,cn)
vector is
Let
i=
r
c
cj
(i=l ...,n)
r
ri+
ri_
cn
cj_l Cj+l
(J=l
,n)
and
X(ilj)
where
denotes the matrix obtained from X by deleting the row i and column
-
(A)
called a -maximizing matrix on Kn if
n
In [3], the following results are proved.
matrix A
K
E
is
If A
THEOREM A.
[aij]
lj()
=
(A)
i_f alj
(A)
if
lf,
for
every @-
i,J=l,.**,n, then
Jn
is the
un_lque
We see that (A)
column
then
@(A)
>
sum
>
n
0
aij
maximizing matrix A
(Cl,...,Cn),
o_n
maximizing matrix on
For A c K
n
if
Now, for A e K with (A)
O.
A
0
)0 for all A c K
vector
K
maximizing matrix on Kn, then
is a
THEOREM B.
and
J.
(X) for all X
)
n
Kn.
with row sum vector (r ,...,r
rl..rn[a ,>
Let A
0
either
>
O,
@(A) for all
ij(A)
Kn,
or
Cl...c n
n
O,
n matrix
denote the n
i
>
defined by
,
lj (A)
aij
For A
K
E
by A) if
function.
say
we
n
tr(AA*)
Let
)
(A)
that A
,n).
(i,J
(A)"
E
K
n
with
(A)
>
0 is A-admlssable (or
A denotes the transpose of A and
n where
tr
A
is
admlssable
denotes
the
trace
denotes the set of all A-admlssable matrices.
It follows from Theorem A that every @-maximizing matrix A is self-admissable
i..
A
(A).
each @-maximlzing matrix A there exists a positive matrix A c Kn such
then the Dittert’s conjecture is true (See section 2).
that A
(A) for
In such a point of view, it would be interesting to study the classes
is
most likely
Such a matrix A should be one which
some particular matrices A c K
If
for
E(A),
n
to posess the property that all -maximlzlng matrices on K n are A-admlssable.
-
In this paper we find some matrices
indiA}for
certain
properties of the Dittert’s function related to the class
2.
THE CLASS
(h)
AND
A’s and investigate some
(A).
MAXIMIZING MATRICES.
From now on let Max(K n
denote the set of all
O--aximizing matrices on Kn.
THE DITTERT’S FUNCTION ON A SET OF NONNEGATIVE MATRICES
THEOREM
2.1.
Max(Kn
then
PROOF.
A e
.(A).
Max(Kn
each A
If
in
matrix
Kn,
{Jn }’ I.e. the D[ttert’s contecture holds.
Let A e Max(Kn and let A-be a positive matrix such that
[Xtj]
. Kn
.
Then
n
n
n
(tr(ATA*)
>
Xi
i-I j=l
by Theorem A.
j
.
CtA
(A)
n
----(
n
Z Xlj
(2.1)
n
i-1 j-1
Therefore the Inequalttltes in (2.1) are all equalities and hence
(A) for all
ij(A)
by__a__ poslt[ve
admlssable
is
711
1,2,...,n
i,j
A is
since
a
positive
matrix.
Now the
assertion of the theorem follows from Theorem B.
Kn with
For A e
Let A
[aij
.
poslt
I
sum
vector
(rl,...,rn)
column sum vector
(cl,...,Cn),
In particular if A e Max(ln)
then A is a
and
n atrix defined by
r
Co
-t_
alj
r
i*l
iivl matrix
(i,j
n
n
n
Since
row
denote the n
n2
ic j
since r
l,...,n).
we see that
t
>
0,
cj >
E
K
n
0 for all i,J
l,...,n because perA
>
0 [2].
We believe that every A e Max(K
is A-admtssabIe, which we can not prove yet.
n
We may ask which matrices A e K are -admisable and which are not. We have an answer
n
to this question.
THEOREM 2.2.
l__f A
is positive semldefinite symmetric matrix In Kn, then A
l__s A-
admlssable.
PROOF.
A(i*I
Let & be a p.s.d, symetric matrix in Kn and let r i be the t-th row sum of
Then the condition that A is
is equivalent to
-admlssable
,n).
n
n
Z Z
i-l j=l
Let
rffirl...r n
and let r
n
rlrj @ij (A)
>n 2
rl...ri_Iri+l...r n (i=l,...,n).
i
n
Z Z
i*l j=l
r
(2.2)
irj
eli
(l)"
.
n
n
i=l
n
n
Z rlrj[r--i+ r--j perA(IlJ)
Z Z
2n2r
Since
n
[(r
n
+
i
perA
by a theorem of Marcus and Merrls [4], we have
rj)r
n
Z Z
n
I I rlr j perA(ilJ) n 2
Then
rIrj
rirj perA(ilJ)]
perA(iIJ).
.
712
SUK GEUN HWANG, MUN-GU SOHN AND SI-JU KIM
n
n
2n2r-
rir j ij(A)
i=l
i=l
2
n
n2(A)
perA
and the proof is complete.
Note that not every atrix A
in K
<
if 0
Ax-admissable
2 is not
K
x
< I/2.
For n
The Kn
and (r 1,...,r
n
so that T
3.
2
(Tn)
n
THE CLASS
.
0
(I,...,1), (c
n)
c
n
n
@(T n)
iffil j=l
tic J ij (Tn)
and hence that Tn is not
(Jn
n matrix.
0
0
0
Then
3, we have an
Let T n denote the following n
EXAMPLE 2.1.
For n=2, the matrix
is A-admissable.
n
2
n-2
n-2
n_--zT).
(2, nl,...,
n)
(n-l)!
>
(n_l)n_2
We have
0
Tn-admissable.
AND THE MONOTONICITY OF THE DITTERT’S FUNCTION.
Another candidate for positive A
nonnegatlve square matrix is called a
column sums are equal to
I.
K
n
(A)
wlth "good"
doub!
is the matrix
Jn"
A
stochastic matrix if all the row sums and
It is conjectured that every n
n doubly stochastic
Here
matrix Is Jn-admlssable (Dokovlc [5] and Minc [6]) but this still remains open.
)) if and only If
we have to notice that A is Jn-admlssable (i.e. A
E(Jn
n
We
show
can
(Jn)
that
.(Jn
n
Kn for n
3 (see
Example
3.1).
However
it
seems
n Identity matrix I n are
K
in
matrices
admlssable. We can show that all diagonal
n are also
THEOREM 3.1. Every diagonal matrix in Kn i__s
that lx(K
n)
).
It is clear that
Jn
and the n
Jn-admlssable.
Jn-admfssable.
PROOF.
Let A
diag(al,...,an
ai+l...a n (i=l,...,n).
@(A)=a
and
n
If a=0,
n
Kn,
a
a
a n and a
n
n
i
al...ai_
Suppose a
there is nothing to prove.
n
>
0.
Then
Jn-
THE DITTERT’S FUNCTION ON A SET OF NONNEGATIVE MATRICES
713
n
(2n-l)a
i-l
Therefore,
n
(A)
<
n
-
t=1
n
> n(2n-l)a.
a--
eli(A)
J=l
if n )2, and the proof is complete.
The Dittert’s function
monotone
the
on
in’(Jn)’r
segment lles
(X)
A(X)
has some nice behavior on the set
segment
line
n
A=[aij]
e
Kn have
For a real number t, 0
vector
(
t
(
Letting
r(t)
c(t)
>
rl(t
c.(t)
c
O, that
d-- r(t)
d
c(t)
[aid(t)]
(rl(t),...,rn(t)) and (cl(t),...,Cn(t))
d
dt
ri_l(t)ri+l(t).., rn(t)
cj -I (t)cj +I (t)
[
c
n(t),
(i=l,...,n),
(j=l,...,n),
Ti(t)}
{r(t)
i;1
n
n
{r(t)
n
i=l j:l
{c(t)
c](t)}
ri(t)},
n
perAt
n
(At
n
n
[ cj(t)},
{c(t)
12
n
l,...,cn).
and let the row sum
n
-d
and column sum vector (c
(t)
c
(t)
-
n)
r (t)
n
l(t)
ri(t
is
line
n
(l-t)J n + tA:
I, let A t
rl(t)
c
’
the
X e K
and the column sum vector of A t be
we compute, for t
namely that
n
row sum vector (r l,...,r
respectively.
so that
Joining
[ [ ij(X)
i=lj=l
n
Let
Jn
(Jn
and A e, (J) whenever
n
function
define by
To show this, let A be a
straight
n
j=l
I_
2
{perAt
n
{r(t) + c(t)
n
n
n
perA
n
i-
[r (t) +
i=I
perAt i lj
i=l
.(t)] perAt(ilj)]}
SUK GEUN HWANG, MUN-GU SOHN AND SI-JU KIM
714
-
which is
d
Kn.
Let A
n
n
n
(At)
Thus we have the following
THEOREM 3.2.
n
A(At).
__If
At
C(J n)_ for all t, 0
I, then the Dirrert’s
t
function is monotone decreasing on the straight llne segment from
It is not hard to show that, for any A
2
2
2
2
Jn
to A.
K2,
3
i-l
On the other hand, the validity of Dlttert’s conjecture for n-2 gives us that
3
Therefore
th@t
Kn
it
follows
(Jn
that K
)
@(&).
2
-C(J2).
However
EXAMPLE 3. I.
does
not
hold
in
Let
O
n
n
n+---
-"
n_
0
n
n
n+l
n+1
0
0
n>4
0
n+---"
0
n+1
n
n+---
0
and let
o
U
3
1/4
-
(b(U n)
4
n xn
3
3
0
0
0
0
Then
and
it
C(Jn )"
( )n
general
THE DITTERT’S FUNCTION ON A SET OF NONNEGATIVE MATRICES
n
3... 3
3
3
3
4
4
3
3
3
3
3
3
3
3
3
3
3
3
n/l
U*n
Hence
2
4n
n
i=1 j=l
eli (Un)
-. -
(sum of entries of U
n n
4(---T)
(n1) n-1
Thus we have
n
2
n+l
[2
2
(4n
n
(U n)
(4n
2
+ 3(6n-10)]
6n + 8).
lj(Un
3
"-T-
4n
2
+ 6n
8)
n-1
-(2n 2
(n+l) n
n
4(n-3)
n
n
i=i j=1
(n_.T)n-1
2n- 8),
which is positive for all n
4.
715
3, telling us that Un is not
Jn-admissable.
CONCLUDING REMARKS.
If, for every A E
we could find a positive matrix A
K such that A is
n
admlssable by A, it would prove the Dittert’s conjecture by Theorem 2.1. It seem to
us that the matrices A or
are two of the strongest candidates for such matrices.
However we may not expect to have a positive matrix
e Z such that all the matrices
n
in Kn are A-admissable.
Max(Kn)
Jn
^
We shall close our discussion here by giving some further research problems.
PROBLEM 4.1.
Determine whether there exists a positive matrix A
K admitting
n
all matrices in KnWe conjecture that such a matrix does not exist.
It
is
admlssable
proved
[4],
increasing on
that
from
every
which
the straight
it
line
p.s.d,
doubly
symmetric
follows
sequment
that
from
the
Jn
stochastic
permanent
to
matrix
function
any p.s.d,
is
is
Jn-
monotone
symmetric doubly
stochastic matrix (Hwang [7]).
PROBLEM 4.1.
admlssable.
Determine
whether every p.s.d,
symmetric
matrix
in
Kn is
Jn-
716
SUK GEUN HWANG, MUN-GU SOHN AND SI-JU KIM
symmetric matrix in Kn is Jn-admlssable, then it follows from
Theorem 3.2 that the Dlttert’s function is monotone decreasing on the straight llne
If every p.s.d,
segment from
Jn
to any p.s.d, symmetric matrix in Kn.
We conjecture that the Problem
4.1 will have an affirmative answer.
Is every -maxlmlzlng matrix A on Kn A-admlssable or
PROBLEM 4.3.
If
4.3
Problem
has
an
a
affirmative
answer,
it
would
prove
Jn-admlssable?
the
Dlttert’s
conjecture as we stated earlier.
Research supported by a grant from the Korean Ministry of Eduatlon
ACKNOWLEDGEMENT.
Inst., 1988.
via the Kyungpook University Basic Science Res.
REFERENCES
I.
MINC, H.
2.
SINKHORN, R.
Theory of permanents 1982-1985, Lin. Multilln.
Multilln.
AI.
12 (1982).
A problem related to the van der Waerden permanent theorem, Lin.
16(1984), 167-173.
AI.
AI.
Appl. 95 (1987), 161-169.
3:
HWANG, S.G.
4.
MARCUS, M. and MERRIS, R. A relation between permanental and determlnantal
adJolnts, J. Austral. Math. Soc. 15 (1973), 270-271.
5.
DOKOVIC, D.Z.
On a conjecture of E. Dlttert, Lln.
On a conjecture by van der Waerden, Mat. Vesnlk 19(4) (1967, 272-
276.
Theory of permanents 1978-1981, Lin. Multllln. Alg. 12 (1982), 227-263.
6.
MINC, H.
7.
8.
On the monotonlclty of the permanent, to appear in Proc. Amer.
Math. Soc.
MINC, H. Permanents, Encyclopedia of Math. and Its Appl. 6, Addlson-Wesley, 1978.
9.
HWANG, S.G.
HWANG, S.G.
A note on a conjecture on permanents, Lln.
44.
The present address of Dr. Suk Geun Hwang is:
Department of Mathematics
Sung Kyun Kwan University
Suwon 440-746
Republic of Korea
AI.
Appl. 76 (1986), 31-