Chapter 1 LINEAR COMPLEMENTARITY PROBLEM, ITS ...

12 Chapter 1. Linear Complementarity Problem, Its Geometry, and Applications
8
>:
fi1i1 ::: fi1ir
.
firi1 ::: firir
This submatrix is known as the principal submatrix of F determined by the subset
fi1:::irg. Denoting the subset fi1:::irg by J, we denote this principal submatrix
by the symbol FJJ. It is (fij : i 2 Jj 2 J). The determinant of this principal
submatrix is called the principal subdeterminant ofF determined by the subset J. The
principal submatrix of F determined by , the empty set, is the empty matrix which
has no entries. Its determinant is de ned by convention to be equal to 1. The principal
submatrix of F determined by f1:::ng is F itself. The principal submatrices of F
determined by nonempty subsets of f1:::ng are nonempty principal submatrices
of F . Since the number of distinct nonempty subsets of f1:::ng is 2n ; 1, there are
2n ;1 nonempty principal submatrices of F . The principal submatrices of F determined
by proper subsets of f1:::ng are known as proper principal submatrices of F .
So each proper principal submatrix of F is of order <
=
n ; 1.
Example 1.2
Let
F =
8
>:
..
9
> :
9
0 ;1 2
1 3 4>
1 5 ;3
:
0 2
The principal submatrix corresponding to the subset f1 3g is >. The princi-
1 ;3
pal submatrix corresponding to the subset f2g is 3, the second element in the principal
diagonal of F .
Several results useful in studying P(S)D matrices will now be discussed.
Results on P(S)D Matrices
Result 1.1 If B = (b11) is a matrix of order 1 1, it is PD i b11 > 0, and it is
PSD i b11 > = 0.
Proof. Let y =(y1) 2 R 1 . Then y T By = b11y 2 1 . So y T By > 0 for all y 2 R 1 , y 6= 0,
i b11 > 0, and hence B is PD i b11 > 0. Also y T By > = 0 for all y 2 R 1 , i b11 > = 0,
and hence B is PSD i b11 > = 0.
Result 1.2 If F is a PD matrix all its principal submatrices must also be PD.
Proof. Consider the principal submatrix, G, generated by the subset f1 2g.
G =
8
>: f11 f12
f21 f22
9
> : Let t =
8
>: y1
y2
9
> :
8
>:
9