On families of anticommuting matrices

Let e1, . . . , ek be complex n× n matrices such that eiej = −ejei whenever i 6= j. We conjecture that • rk(e21) + rk(e 2 2) + · · ·+ rk(e 2 k) ≤ O(n log n). We show that (i). rk(en1 ) + rk(e n 2 ) + · · ·+ rk(e n k ) ≤ O(n log n), (ii). if e21, . . . , e 2 k 6= 0 then k ≤ O(n), (iii). if e1, . . . , ek have full rank, or at least n−O(n/ log n), then k = O(log n). (i) implies that the conjecture holds if e21, . . . , e 2 k are diagonalizable (or if e1, . . . , ek are). (ii) and (iii) show it holds when their rank is sufficiently large or sufficiently small.