2
A theorem of the alternative
The separating hyperplane theorem has a variety of applications. Amongst them is the very
interesting result about existence of solutions to linear systems which we can use to determine
conditions when arbitrages cannot exist. Suppose
A
is an
n
×
m
matrix and for
z
∈
R
k
write
z >
0 when
z
j
>
0 for each
j
and
z
≥
0 when
z
j
≥
0 for each
j
. Recall that (
Mz
)
T
=
z
T
M
T
.
Theorem 2.1 (Gordan’s theorem)
Exactly one of the following systems has a solution:
(1)
y
T
A >
0
for some
y
∈
R
n
;
(2)
Ax
= 0
,
x
≥
0
for some nonzero
x
∈
R
m
.
Proof:
start by showing by contradiction that the two systems cannot both have solutions.
If each has a solution then
z
T
=
y
T
A >
0 and hence 0 =
y
T
(
Ax
) =
z
T
x >
0 which is
impossible.
Next suppose that system (1) has no solution i.e. the two nonempty convex sets
S
1
=
{
z
∈
R
m
:
z
=
A
T
y, y
∈
R
n
}
,
S
2
=
{
z
∈
R
m
:
z >
0
}
are disjoint, that is
S
1
∩
S
2
=
∅
(though clearly
z
= 0 is on the boundary of both sets).
Hence there exists a vector
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This note was uploaded on 02/04/2011 for the course MATH 3301 taught by Professor Dri.m.macphee during the Spring '10 term at Durham.
 Spring '10
 DrI.M.MacPhee
 Math, Linear Systems

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