Page 429 - Advanced Linear Algebra

Positive Solutions to Linear Systems: Convexity and Separation 413

positive orthant, must have negative slope. Our task is to generalize this to
:
s .

This will lead us to the following results, which are quite intuitive in s and s :

:q s ž bb £ J ¯ : q s b ~ J (15.1)
and

:q s ž b £ J ¯ : q s b b ~ J (15.2)
Let us translate these statements into the language of the matrix equation
(% ~ . If : ~ RowSpace ²(³, then : ~ ker ²(³ and so we have
ž
ker²(³ q s bb £ J ¯ RowSpace ²(³ q s b ~ J
and

ker²(³ q s b £ J ¯ RowSpace ²(³ q s b b ~ J
Now,

RowSpace²(³ q s b ~ ¸#( “ #( € ¹
and
RowSpace²(³ q s bb ~ ¸#( “ #( ˆ ¹

and so these statements become
(% ~ has a strongly positive solution ¯ ¸#( “#( € ¹ ~J
and

(% ~ has a strictly positive solution ¯ ¸#( “#( ˆ ¹ ~J
We can rephrase these results in the form of a theorem of the alternative , that
is, a theorem that says that exactly one of two conditions holds.

Theorem 15.1 Let ( C ² s Á . ³
1 Exactly one of the following holds:
)
)
a (" ~ for some strongly positive "  s .
b #( € for some #  s .
)
2 Exactly one of the following holds:
)
a (" ~ for some strictly positive "  s .
)
)
b #( ˆ for some #  s .
Before proving Theorem 15.1, we require some background.
Convex, Closed and Compact Sets
We shall need the following concepts.