Diagonalization in Jordan pairs

Abstract

As every student of linear algebra knows, a rectangular matrix over a division ring can be diagonalized by elementary row and column operations. Similarly, there are normal forms for alternating and hermitian matrices which can be achieved by simultaneous row and column operations. Since matrices of this kind form the main examples of Jordan pairs, it is natural to ask whether similar results hold in the Jordan setting. We show that this is in fact the case, study the obstruction to diagonalizability, the defect, and also prove that nondegenerate Jordan pairs admit a rank function sharing many properties with classical matrix rank. Let V= (V+, V) be a Jordan pair and S = {e,, . . . . e,} a set of orthogonal idempotents. The S-diagonal elements are those in xi= 1 V,(e,). Now suppose that V is nondegenerate and satisfies dcc on principal inner ideals. An element x E V” (a = i) is called diagonalizabfe if x is S-diagonal for some S consisting of division idempotents. To see that, for matrices, this is the same as the usual notion of diagonalizability, suppose that V has in addition act on principal inner ideals. Then V contains a frame, that is, a finite set F of orthogonal division idempotents such that V,,(F) = 0. For rectangular or hermitian matrices over division rings, F can be taken to consist of the diagonal matrix units (e,? = Eii) whereas for alternating matrices over a field, e,? = Eli1,2iE,,,zi-, . The Jordan analogues of elementary row and column operations are the inner automorphisms B( V:(e), V;(e)) and /?( V:(e), V,(e)) (where e E F) which generate the group of F-elementary automorphisms of V [6]. Any set of orthogonal division idempotents can be transformed (up to association) into F by an F-elementary automorphism [6, Th. 21. It follows that x is diagonalizable if and only if cp +(x) is F-diagonal for some elementary automorphism cp of V. This shows the equivalence with the usual definition.