On Families of Anticommuting Matrices.

Let e1,..., ek be complex n × n matrices such that ei ej = −ej ei whenever i ̸= j. We conjecture that rk(e21 ) + rk(e22 ) + · · · + rk(e2k ) ≤ O(n log n). We show that: (i). rk(en1 ) + rk(en2 ) + · · · + rk(enk ) ≤ O(n log n), (ii). if e21,..., e2k ̸= 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,..., e2k are diagonalizable (or if e1,..., ek are). (ii) and (iii) show it holds when their rank is sufficiently large or sufficiently small.

Authors

• Pavel Hrubes

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