Some algebraic properties Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol67.Issue3.Jun2000:

K . Neusser Economics Letters 67 2000 273 –281 279

3. Some algebraic properties

n 3n From now on I view the set of all n 3 n matrices, R , as a semigroup with respect to the matrix 3 n 3 n n 3n multiplication. The identity element in R is I . For any given E [ R consider the subset of n E n 3 n R defined in the following way: n 3n u 5 h A AE 5 EA 5 A and B [ R : AB 5 BA 5 E j E n 3n 2 If is nonempty, there exists A [ and B [ R such that AB 5 E. Thus E 5 EAB 5 AB 5 E so E E that E is idempotent and E [ . It is also straightforward to see that is closed under E E n 3n multiplication and that EBE [ is the inverse of A in . In other words is a subgroup of R E E E with identity element given by E. The subgroup is maximal because every subgroup containing E is in fact a subgroup of . This can be seen by noting that E must be the identity of any group in E which it lies, so that any A[ satisfies the defining conditions for Drazin, 1958: 513. This leads E 4 to a family of disjoint maximal subgroups h j where E runs over all idempotent matrices. This E n 3 n family does not form a partition of R because, as is shown in Theorem 4, matrices with index strictly greater than one do not belong to any subgroup. Theorem 4. Under Assumption 1, Ind P 5 1 if and only if P belongs to some subgroup. g Proof. ⇐ : Suppose P belongs to some subgroup . Then P must have an inverse P in . This g g g g g g g [ inverse satisfies PP 5 P P, P PP 5 P , PP P 5 P. Therefore P is nothing but P , the 5 group inverse of P. Therefore IndP 51, given Assumption 1. n 3n r 3r ⇒ : If Ind P 51, there exist invertible matrices P [ R and I 2 J [ R such that r n 2r 21 P 5 P P S D I 2 J r Define the set of matrices as [ PP 21 r 3r n 2r u 5 P P M [ R , rank M 5 r j . [ s d H S D PP M It is easy to see that P [ and that is a group. h [ [ PP PP Theorem 5. 5 [ [ PP PP [ Proof. Any element in satisfies the definition of . Moreover, PP is idempotent and [ [ PP PP [ PP [ , thus is a subgroup of . [ [ [ PP PP PP 3 A semigroup is just a set together with an associative binary operation. 4 The fact that the groups are disjoint can be established as follows. Let G and G be two subgroups such that E ± F and E F G G ±[. Then for any A [ G G , there exists B and C such that EF 5 BAF 5 BA 5 E 5 AB 5 FAB 5 FE and E F E F EF 5 EAC 5 AC 5 F 5 CA 5 CAE 5 FE. This implies that F [ G and E [ G . The two groups must therefore be equal E F because they are both maximal subgroups. 5 [ This is the justification of the name ‘group inverse’ for A . 280 K . Neusser Economics Letters 67 2000 273 –281 21 q 3q [ [ If A is any A [ , A 5 Q Q with B [ R and rank B 5 q. Because AA 5 PP [ S D PP B these matrices are similar and therefore r 5 q. Furthermore there exists an invertible matrix R such [ [ that Q 5 PR. Observing that the definition of implies PP A 5 A and APP 5 A, it is easy to [ PP verify that A can be written as 21 A 5 P 21 P S D R BR 22 22 where R is the 2,2-element of the appropriately partitioned matrix R. Thus A [ . h [ 22 PP [ 2 [ 2 Note that the set of matrices hE,P,P ,P ,P , . . . j constitutes a commutative Abelian [ [ subgroup of . The identity is again the matrix E 5 P P 5 PP . The subgroup is cyclic and E generated by the positive and negative powers of P where negative powers are interpreted as powers [ of P . Following Grillet 1995: 27, we introduce a partial order on the set of idempotent matrices: Theorem 6. Partial order: The binary relation defined by E F ⇔ EF 5 FE 5 E represents a partial order on the set of all idempotent matrices. Proof. The reflexivity follows from E being idempotent. Symmetry is also immediate from the definition. Finally, if E F and F G then EG 5 EFG 5 EF 5 E 5 FE 5 GFE 5 GE. Thus E G and is also transitive. Let us translate this definition into our context. The definition of E F implies that E and F commute. Moreover, E and F are diagonizable matrices, consequently they must share the same 21 eigenvectors Strang, 1988: 259. E and F are therefore both of the form P P . They may S D I r differ only with respect to r. Thus for any two idempotent matrices E and F, E F is equivalent to E and F share the same eigenvectors and r 5rank E r 5rank F. The cointegration space E F corresponding to E is therefore included in the one corresponding to F.

4. Conclusion