Partially Ordered Group of the 2×2 Symmetric Matrices

This paper present results partially ordered group on the set of 2 × 2 symmetric matrix through its positive cones. There are two positive cones that is the subset of the symmetric matrix. The first cone is constructed by defining positive matrix as the matrix that each entry of the matrix is positive. The second cone is constructed by using the characterization of the matrix.


INTRODUCTION
In this paper, we consider the set of 2 × 2 matrix over real number with the addition operation.It can be shown that the set   R S 2 is also group.
Let two elements A and B in symmetric matrix   R S 2 .Now we define the order of the two elements by T [1].By the defined order, the 2 constructs partially ordered group.Moreover, a set P is said to be totally ordered set if it is partially ordered set that satisfies y x  or x y  for all elements x and y in P .For instance, the real number set R with Binary relation "  ".It is the partially ordered set since for any two real numbers x and y , y x  or x y  .It is always be ordered.Then, let us define the Cartesian product of the real number set R as follow: . By defined the order as It is also partially ordered set but it is not totally ordered set since there exists (1,4) and (2,3) on R R  that is not satisfies whether (1,4) ≤ (2,3) or (1,4) ≥ (2,3).

Inner Product
Let x and y are two elements of n dimensional R , n R .The standard inner product of R is a function that is defined by with T y is a transpose of y .The properties of the standard inner product for any elements , , y x and z in n R and the scalar  in R as follows [4] [5]: (1) Symmetric: ,  .
(2) Positive definite: if and only if 0  x .
(3) Linearity of the addition: (4) Linearity of multiplication: Recall the set of 2 × 2 symmetric matrix over R , . By the inner product of matrix defined Since the symmetric matrix has the property A A T  .The following theorem gives the characteristics of symmetric matrix.

Partially Ordered Group
A Partially ordered set G which is also a group is said to be partially ordered group if the order of G is preserved under addition that is if x y  then a x a y    and x a y a    for any elements y x, in G and the scalar a in R .Define the positive cone  G by all positive elements of , and the negative cone Any group can be constructed as partially ordered group with the positive cone.
Theorem 2 Let P is the subset of group G under addition that satisfies two properties: (1) For any two element a and b of P , b a  is element of P .
(2) The intersection of the set P and its negative, P  , is identity element 0 .Then the set G can be constructed as partially ordered group by defined b a  if and only if P b a   Moreover, this results the group G to be partially ordered group with the positive cone P .We denote the partially ordered group with its cone by   P G, .

RESULT AND DISCUSSION
Recall again the set of 2 × 2 symmetric matrices that is group under matrix addition operation.The following present two positive cones constructed by the subset of   R S 2 .
Lemma 1 with 0 P is partial ordered set with order Proof: It will be shown that the set   R S 2 with the 0 P is partially ordered.

For any two elements
is an element of the intersection of 0 P and its negative . By definition, , , v u and w are elements of R and are positive.Thus, , , v u   and w  are negative.In this case, the elements , , v u and w of R that satisfy the condition is zero.defined by [7]:

Based on the Theorem 2, by the order
with 1 P is partial ordered set with order: Proof: It will be shown that the set   R S 2 with the 1 P construct partial order.

For any two elements
. By the properties of the inner product on   Therefore, 2. Let A is an element of the intersection of 1 P and its negative.By definition, A is an element of , respectively, to be partially ordered groups.

A
under addition operation matrix.The matrix over real number  for any two matrices A and B that satisfies i There exists an identity element e in G and for all a element in G satisfies For all A and B are elements of the 2 × 2 symmetric matrix   For all A and B are elements of   1P and A is an element of