I ntrn. J. Math. Math. Sci.
Vol. 5 No. 2 (1980) 295-504
293
RANKED SOLUTIONS OF THE MATRIC EQUATION A
1X
A2X 2
A.Duane Porter
Mathematics Department
University of Wyoming
Laramie, Wyoming 82070
Nick Mousouris
Mathematics Department
Humboldt State University
Arcata, California 95521
(Received December 8, 1977 and in Revised form February 20, 1979)
GF(p z)
ABSTRACT. Let
s x m
of rank
r
I
pZ
denote the finite field of
A2
and
be
of rank
n
s
r
2
elements.
AIX1
kl,
and
X
2
is
n
ber of solutions
AIX1
X
n
A2Y 1
of rank
t
XI,
Xn,YI
Y
k
2.
A2X2,
A 1 be
with elements from
XI,X 2
In this paper, formulas are given for finding the number of
which satisfy the matric equation
Let
where
X
1
is
GF(pZ).
over GF(p
m x t
z)
of rank
These results are then used to find the num-
’Ym’m’n >
i,
of the matric equation
m
KEY WORDS AND PHRASES. Finite Field, Matric equation, Ranked Solutions.
1980 MATHEMATICS SUBJECT CLASSIFICATION
CODES:
15A 24.
294
i.
A. DUANE PORTER AND N. MOUSOURIS
GF(q)
INTRODUCTION. Let
GF(q)
Matrices with elements from
odd.
A(n,s)
B
will denote a matrix of
having
I
l(n,s;r)
r, and
z
rows and
n
A,
A(n,s;r)
columns, and
s
I
denotes a matrix of
n
r
p
elements,
will be denoted by Roman capitals
will denote a matrix of the same dimensions with rank r.
matrix of order
p
q
denote the finite field with
denotes the identity
rows and
s
columns
in its upper left hand corner and zeros elsewhere.
r
In this paper we find the number of solutions
Xl(m,t;kl) X2(n,t;k2)
to the
matric equation
AI
where
AI
Al(S,m;r I)
and A
we call this the ranked case.
the unranked case.
solutions
A2(s,n;r2).
2
(i.i)
A2X2,
Since the ranks of
XI,X2
are specified,
If the ranks were not specified, we would call it
In section 4 we apply this result to find the number of
XI,...,Xn,YI,...,Ym,
m,n e i, to the matric equation
AIXI
Ym’
Xn A2YI
(1.2)
both in the unranked and ranked cases.
Equation (i.i) is a special case of the matric equation
+
AIXI +
AnXn
(1.3)
B,
and equation (1.2) is a special case of the more general equation
AIXII
Porter
case.
+
Xlm(1)
+
[ 6] found the number of solutions
AnXnl
Xnm(n)
,Xn
X
I
to
(1.4)
B.
(1.3) in the unranked
We could find the number of solutions to (1.3) in the ranked case if we
could find the number of ranked solutions to
ed solutions to
solutions to
AIX1 + A2X2
X1
number of unranked
X
n
B
B
AIXI + A2X2
B.
The number of rank-
together with the formulae for the number of
would give the number of solutions to (1.4).
solutions to
X
1
Xn
B
is given by Porter in
5].
The
The
RANKED SOLUTIONS OF THE MATRIC EQUATION
X
X
number of ranked solutions to
I
appears in [ 9], by the authors.
B
n
295
Presently the authors know of no published results, unranked or ranked glving the number of solutions to (1.4) except when (1.4) reduces to (1.3).
There
are partial results in the unranked case to the analogous problem
UIA + BVI
Ua
b
a
with
I] found the number of unranked solutions
Hodges
found partial results in the unranked
2] and Porter [?
Hodges
i.
case when
C.
Vb
are arbitrary and Hodges [ 3] discussed ranked solutions when
a,b
a=b=l.
NOTATION AND PRELIMINARIES.
2.
g(m,t;s) of
the number
m
t
matrices of rank
s
g(m,t;s)
s(s-l)/2 H
q
4] gives
A well known formula due to Landsberg
m-i+l
over
s
l)q t-i+l
(q
GF(q).
l)/(q i
(2.1)
i),
i=l
for i< s < min(m,t), g(m,t;0)
Y
G(e,t,m;k,s)
q
s (e-m)
g(e
B, where
AX
A(s,m;0)
A
X
has
(2.2)
s;k -s).
m,t
In 9,Theorem 3 ] the authors found the number of solutions
matric equation
U(m,t;s) and
[11] to be
is given by Porter and Riveland
k
rank
s < 0.
< s or
can be chosen such that
Y
m,t), then the number of ways that
Y(e
mln(m,t)
is fixed, U
col[U,Y], where U
X
And
X(e,t)
X
If
for
0
I, and g(m,t;s)
and
B(s,f;).
B
X(m,f;k) to the
This number is
given by
N(A,B,k)
h(Bo)
where
l(s m;p)
PAQ
h(B
by
0
O
h(Bo) q
(m-0)8
is defined as follows.
then
otherwise.
B
PB
o
(
ij
If
and
h(BO )L(m,f;p,8 k)
B;k- )
g(m- O f
P,Q
(2.3)
are nonsingular matrices such that
h(Bo)
The number of solutions, when
i if
ij
0 for
i >
,
there are any, is denoted
L(m,f;0,8,k).
Let
A
A(n,s;r)
and
B =’B(n,t;u).
Then Porter [5] showed that the number
296
A. DUANE PORTER AND N. MOUSOURIS
Xl(S,Sl) Xi(Si_l,S i)
of solution
A
equation
X
B. when there are solutions
N(a,s,t,si,r,u
t(Sa_l-r)+ssl+SlS2+
q
Xa(Sa_l,t)
1 <i < a,
for
to the matric
is given by
+s a
imin(, t)H(r,t,Y;Za_l)
2Sa
Za_l =0
-Za_ISa_ I
q
min
a_l
g(m,t;s)
.
is given by
qUZ
H(s,t,u,z)
g (Za_i+
I
Za_l=0
i=2
where
(2.4)
(Za_i+I Sa-i+I)
z
(2.1), and
H(s,t,u;z)
(_l)Jq j (j -2u-i)/2
Sa-i+1 Za_ i) q
-Za_iSa_ i
is given in [i] to be
g(s
u,t-u;z
j),
j=0
where the bracket denotes the q-binomial ceefficient defined for non-negative
intergers by
E]
if
j > w.
i,
j-i
]
q_l) / (l_qi+l)
H (i
1 < i < u,
if
i=0
For the purposes of this paper we take
there will always be solutions to
X
Xa
A
I
s
in (2.4).
By [5]
B, and this number can
be repre-
sented by
Ma(S,Sl,...,Sa_l,t,u)
The number of matrices
D
1
Dl(d,b)
K(a,b,c,d)
and
q
D
D2(a
2
(c-d)d
g(d,b + d
Tn(do,...,dn;kl,...,kn,8)
matric equation
D
...Xn
=I
(2.5)
such that
d,b;c -d), d < min(a,c)
c;d)g(a
where
for
for
D(a,b;c)
d,b;c
d).
D
col[Di,D2]
where
is given in [I0] to be
The number
Xl(d o,dl;kl),...,Xn(dn_ l,dn;kn)
of solutions
B
(a’s’t’si’s’u)
B
B(d o ,dn;8)
to the
is given in [9] by the
RANKED SOLUTIONS OF THE MATRIC EQUATION
297
following three formulae:
T
T
0
if
k
B,
i
if
k
8,
l(do,dl;kl,8)
2(dO,dl,d 2, ;kl,k2, 8)
K(d
.
r
T (d
n
k ,8)
n
,dn;kl,
o
n
i =8
n
H
m=3
where n -> 3,
3.
I
..... ,kj_l)
rj
min(k
I
for
d
I
k
I
I
8)L(dl,d 2
r
rn-i
i
n
;k
l=in
k
I 8 2)
(2.6)
3
i3=i4
r 2(do,di,kl,k2,i3 )"
K(do dm-l’ Im" im+l)L (dm-l’ dm’ +/-m’ im+l’km)
3,
j
n
and
8.
in+I
THE MAIN RESULT.
THEOREM i.
X2(s,t;k2)
for
If
A(s,m;p),
A
p,k
I
> k
2,
then the number of solutions
to the matric equation
AX
is given by
N(m,t;0,kl,k2)
uated using (2.1) and
PROOF:
Let
P,Q
I
(3. I)
X2,
g(0,t;k2)L(m,t;o,k2,kl),
L(m,t;p,k2,kl)
be
Xl(m,t;kl).
nonsinular
where
is evaluated using
matrices such that
g(0,t;k 2) is eval-
(2.3).
PAQ
l(s,m;0).
Then
(3.1) can be rewritten as
l(s,m;)Q-IxI
PX2.
(3.2)
A. DUANE PORTER AND N. MOUSOURIS
298
col[Y,0] where Y
The left hand side of (3.2) is of the form
a particular
X
2(s,t;k2)
(m,t;kl)
there will be matrices
the form
col[Y,0] where
X
same number of matrices
rank
k2.
Y(p,t;k2).
Y
The number of
2
t
XI,X2
that
(2.3).
satisfy (3.1)
matrices of rank
g(o,t;k2).
formula (2.1) and denoted by
g(p,t;k2)L(m,t;o,k2,kl),
k
p
X
2
to
(3.1)
matrices of
Landsberg’s
the number of X
I
L(m,t;,k2,k I)
can by represented by
XI,X2
t
is given by
2
For each such
Therefore the number of solutions
and a matrix of
P -i is nonslngular there are the
Since
with this property as there are
0
For
which satisfy
P -I
(3.2), and therefore (3.1), provided X2 is the product of
Y(p,t).
such
as given by
is given by
and the theorem is proved.
It should be noted that Theorem i is a special case of a theorem of Hodges
However, our proof, and so the form of the resulting formula,is quite dlf-
3].
ferent since Hodges uses exponential sums in
his proof and we do not.
Our proof
of Theorem i is consistent with the methods of proof used in the rest of this
paper.
THEORF.M 2.
A
2
PA2Q
A2(s
l(s
Let A
A(s,m;p)
n,m;u 2) wth
n,m;u2)
and
the number of solutions
col
n < s.
Let
AIQ
BI,B2
[AI,A2]
P,Q
where
A
1
Al(n,m;u I)
and
be nonsingular matrices such that
where
X l(m,t;kI), X2(n,t;k2)
B
2
to the
B2(n,m
u2;S).
matric equation
Then
RANKED SOLUTIONS OF THE MATRIC EQUATION
299
(3.3)
for
i
>
k2
is given by
N(m-
g(B,t;k2)
where
PROOF:
For
2,t;B,kl,k2)
is given by
AI,A2
g(B,t;k2)L(m- =2,t;B,k2,kl),
(2.1)
L(m-
and
2,t;B,k2,kl)
is given by
(2.3).
defined as above we can write (3.3) as the system of
equations
A
Substituting
by
2
P
-i
X2,
(3.4)
A2XI
0.
(3.5)
n,m;2) Q
l(s
-i
into
Let
Q
Q-IXI
-iXl
coi[
Yl’ Y2 ]’
in (3.6) by col[Y
X
I
1
2
-IxI
n,m;2) Q
YI YI(2 ’t)
where
(3.6) are that
X
I
obtain
B
l
and
Y2 Y2 (m -2 ’t)"
Replacing
YI
=0, rank
Y2 =kl
and X
I =Qcol[0
2].
in (3.4) gives
AIQ
-AI Q
(3.6)
=0.
we have that necessary and sufficient conditions
to be a solution of
Using this formulation for
Let
(3.5) and multiplying on the left
P we obtain
l(s-
for
AIXI
2]’ where Bl(n,=2
and B
(3.7)
X2.
2
B 2(n,m
2’ )
in
(3.7) we then
A. DUANE PORTER AND N. MOUSOURIS
300
N(m-
By Theorem I there are
Q
Since
(3.8)
X 2.
B2Y 2
2’ t;B’kl’k2)
pairs
Y2,X2
which satisfy (3.8).
is nonsingular there are the same number of pairs
XI,X 2-
which satisfy
(3.3).
(3.4)
Equations
(3.5)
and
equations in the two matrices
systems.
represent a special system of two simultaneous
XI,X2.
Very few results seem to exist for such
The authors are unable to find the number of solutions to the general
system of two simultaneous equations in
to enumerate the ranked solutions to
THEOREM 3.
Let
A
Al(S,m;rl)
I
XI,X2.
AIXI
+
and
A
Such information would allow us
A2X2
2
B.
A2(s,n;r2).
PIAIQI
PI,QI be noncol[A21,A22],
Let
l(s,m;rl). Define PIA2
where
A21 A21(r,n;a I) and A22 A22(s rl,n;a2). Let P2,Q2 be nonsingular
matrices such that P2A22Q2
[B I,B2] where
r l,n;2). Define A21Q2
l(s
B
B2(rl,n- a2;B). Then the number of solutions Xl(m,t;kl) X2(n,t;k2) to
2
singular matrices such that
the matric equation
(3.9)
AIXI A2X2
is given by
N(m,n,t;r l,r 2,k l,k 2,al,i,B)
min(al, kI
I
kll =0
G(m,t,rl;kl,kll)g(B,t;kll)L(n
where
G(m,t,rl;kl,kll
(2.1),
and
PROOF:
L(n-
can be evaluated using
e2,t;B,kll,k2)
The number of solutions to
s2,t;B,kll,k2),
(2.2),
g(B,t;kll)
is given by
is given by (2.3).
(3.9)
is the same as the number of solutions
to
l(s,m;rl)X I PIA2X2
(3.10)
RANKED SOLUTIONS OF THE MATRIC EQUATION
X
Letting
A
(3. I0)
PIA2
Xll(rl,t;kll ), XI2
col[Xll,Xl2], XII
1
301
XI2 (m -r’t;kl2)
and
becomes
(3.11)
AX2
XII,X2
Theorem 2 gives the number of pairs
the number of
Xll(rl,t;kll)
is given by
XI,X2
(2.2), denoted by
X12(m-rl,t;kl2)
G(m,t,rl;kl,kll).
such that
For each
Xl(m,t;k 1) --col[Xll,X12
Therefore the number of solutions
G(m,t,rl;kl,kll)S(,t;kll)L(n-2,t;,kllk
KII where KII < I by the hypothesis of
is given by the product
(3.9)
to
that satisfy (3.11).
summed over the possible values of
Theorem 2.
4.
SOME APPLICATIONS.
We can now use Theorem 3 together with some other known results to find the
X
of solutions
number
Xn’YI’
I
Ym
to (1.2) in both the unranked and
unranked cases.
A2(m,t;r2). Let PI,QI be nonsingular matrices
l(m,So,’rl). Define PIA2 col[ I A22
where
A21 A21(rl,to;U I) and A22 A22(m rl,to;U2). Let P2,Q2 be nonsingular matrices such that P2A22Q2
l(m-rl,to;U2). Define A21Q 2 [B I ,B2],
THEOREM 4.
where
B
2
Let
B 2_(r
A
I
Al(m,So;r I)
such that PIAIQI
and
I
t
u2;_
O
A
2
Then the number of
Xn(Sn_ l,sn), Yl(to,tl),.....,Ym(tm_l,tm),m,n
_> I,
solutions
Sn =tm
to
X l(s
o,s I)
(1.2)
is given
by
min
(So,
il=0
s
n)
min
(to,
t
i2=0
m)
N(So,to,Sn;rl,r2,il,i2,el,e2,8)"
A. DUANE PORTER AND N. MOUSOURIS
302
M (s
n
N(s
where
Mn (s o
s
PROOF:
where
0
to
m
m
i
2)
can be evaluated using (3.9)
(2.5).
can be evaluated using
tm,i 2)
o
t
Consider the equations
U
_< i2<_
V
U(So,Sn,’il),
U
X1...Xn,
(4.2)
V
YI...Ym,
(4.3)
V(to,tm;i 2),
0
_< i I <_ min(So,...,s n)
U
or
V.
M
n
and M
m
of solutions to (1.2)
amd
(4.1) is given by (3..9)
The numbers
Mn(So,...Sn,il)
and (4.3),
can be evaluated using (2.5).
summed over the possible ranks of
n m
to
N(So’to’Sn;rl’r2’il’i2’=l’a2’8)"
respectively, for a fixed
NM M
(4.1),
represent the number of solutions to (4.2)
Mm(to,...,tm,i 2)
The product
AIU A2V
The number of solutions U,V
min(to,...,tm).
and is represented by
and
M (t
and
il)Mm(t
8)
il i2 el 2
Sn;rl r2
nil)
..,s n
o
U
gives the number
and V
in the unranked case.
The next theorem is proved in the same way that Theorem 4 is proved except
that we use
(2 6)
to obtain the number of ranked soltlons
to the matric equations
THEORt 5.
Let
as in Theorem 4.
Xl(So’sl’Jl)’
(4.2)
X
I
Xn,Y I
...,Ym
and (4.3)
A1,A2 ,P1,Q1,P2,Q2,A21,A22,B1,B2,Bll,B12,l,2,
and
Then the number of solutions
’Xn(Sn-l’Sn;Jn) ’Yl(to’tl;kl)’
’Ym(tm-l’tm;km)’m’n
_>
i,
8
be
RANKED SOLUTIONS OF THE MATRIC EQUATION
s
n
t
to
(i. 2)
mln (j
is given by
j
i
n)mln (k I
km)
N(So" to’Sn; rl’ r2’il’ i2’si’2’ 8).
i2--0
il=O
T
where
n(So
n’"j i’
N(So,to,Sn;rl,r2,il,i2,l,e2,8)
Tn(So,...,Sn,’Jl,
303
Jn’i I)
and
j
tm;k I
n’ il)Tm(t
is evaluated using
Tm(to,...,tm;kI,
(3.9)
km,i 2)
km,i 2)
and
are evaluated using
(2.6).
NOTE:
This paper was written while the second named author was on leave at the
University of Wyoming.
REFERENCES
[I] Hodges,John H. Some matrix equations over a finite field, Annali di,
Matematica 44, (1957) 245-250.
Note on some partitions of a rectangular matrix, Accademia
Nazionale Dei Lincei, 8, N9, (1976) 662-666.
[2] Hodges,ohn H.
[3] Hodges,John H.
Ranked partitions of rectangular matrices over finite fields,
Accademia Nazionale Dei Lincei, 8, 60, (1976) 6-12.
[4] Landsberg, Georg. Uber eine Anzohlbestimming und eine angewandte MathematiC,
III, (1893) 87-88.
[5] Porter, Duane A. The matric equation AX1 +...Xa =B
Dei Lincei, 8, 44, (1968) 727-732.
Accademia Nazionale
The matric equation
+ AmXm B in GF(q)
I +
Journal of Natural Sciences and Mathematics,13, i, (1973) 115-124.
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AIX
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A. DUANE PORTER AND N. MOUSOURIS
[7]
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[8]
Porter, Duane A.
[9]
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[i0]
Porter, Duane A., and Nick Mousourls, Ranked solutions to some matrlc
equations, Linear and Multilinea r Algebra 6, (1978) 153-159.
[Ii]
Solvability of the matric equation
and its Applications, i_5, (1978).
AX
B, Linear Algebra
Porter, Duane A., and A. Allan Riveland, A generalized skew equation over
a finite field, Mathematische Nachrlchten, 6__9, (1971) 291-296.