Kato-Taussky-Wielandt commutator relations and characteristic curves

Fergus John Gaines

Vol. 61 (1975), No. 1, 121–128

DOI: 10.2140/pjm.1975.61.121

Abstract

Let A and B be n × n matrices with elements in a field ℱ and let Δ_AB = AB − BA. Let f_k(x) = x^2K+1 − c₁x^2K−1 + c₂x^2K−3 + ⋯ + (−1)^Kc_Kx, where the c_i are in ℱ and K = k(k − 1)∕2. In this paper we examine the consequences of the relation f_k(Δ_A)B = 0, where 1 ≦ k < n, and show how the replacement of A by xA + yB, when k = 2, leads to a splitting of the characteristic curve, det(xA + yB − zI) = 0, into lines and conics.

Mathematical Subject Classification 2000

Primary: 15A21

Milestones

Received: 15 March 1975

Published: 1 November 1975