Page 427 - Advanced Linear Algebra
P. 427
Chapter 15
Positive Solutions to Linear Systems:
Convexity and Separation
It is of interest to determine conditions that guarantee the existence of positive
solutions to homogeneous systems of linear equations
(% ~
where ( C ² s Á . ³
Definition Let # ~ ² ÁÃÁ ³ s .
1 # ) is nonnegative , written # , if
for all
(The term positive is also used for this property. ) The set of all nonnegative
vectors in s is the nonnegative orthant in s À
2 # ) is strictly positive , written # , if is nonnegative but not , that is, if
#
for all and for some
The set s b of all strictly positive vectors in s is the strictly positive
orthant in s À
3 # ) is strongly positive , written # , if
for all
The set s bb of all strongly positive vectors in s is the strongly positive
orthant in s À
We are interested in conditions under which the system (% ~ has strictly
positive or strongly positive solutions. Since the strictly and strongly positive
orthants in s are not subspaces of s , it is difficult to use strictly linear
methods in studying this issue: we must also use geometric methods, in
particular, methods of convexity.
