A FIXED POINT THEOREM FOR VOLUME PRESERVING LINEAR TRANSFORMATIONS
M. Moskowitz · International Journal of Pure and Apllied Mathematics · 2016
In this note we derive consequences of the fact that if g ∈ SL(n), where n ≥ 2, and σi(g) are the coefficients of its characteristic polynomial, then g has 1 as an eigenvalue if and only if n-1 i=1 (-1) i-1 σi(g) = 0 or 2 according to the parity of n.These are: Corollary 2. Let D be a real or complex n × n matrix of tr(D) = 0.If D is singular, then one of the following equations holds (according to the parity of n).Conversely, if D has no eigenvalues in 2πiZ and the appropriate one of these equations holds, D must be singular.If A has only non-negative eigenvalues and one of the following equations holds (according to the parity of n)then [A, X] = 0 for some X ∈ M, not a linear combination of A and I.Corollary 4. Let A ∈ M and have all non-negative eigenvalues.Then the action of Exp(A) on M by conjugation has a fixed point which is not a linear combination of A and I.