Botta - Pierce - Watkins theorem

Let 𝔽 be an arbitrary field, and let n be a positive integer. Consider n(𝔽), the vector spaceMathworldPlanetmath of all n×n matrices over 𝔽. Define

  • 𝔰𝔩n(𝔽)={An(𝔽):tr(A)=0},

  • 𝒩={An(𝔽):Ais nilpotent},

  • 𝒢n(𝔽)={An(𝔽):det(A)0}.

Notice that 𝔰𝔩n(𝔽) is a linear subspace of n(𝔽) and 𝒩𝔰𝔩n(𝔽).

The Botta – Pierce – Watkins theorem on linear preservers of the nilpotent matricesMathworldPlanetmath [BPW] can be formulated as follows.

Theorem 1

Let φ:sln(F)sln(F) be a linear automorphismMathworldPlanetmathPlanetmath. Assume that φ(N)N. Then either PGLn(F)cF{0}Asln(F):φ(A)=cPAP-1, or PGLn(F)cF{0}Asln(F):φ(A)=cPATP-1.

The original proof is based on the Gerstenhaber - Serezhkin theorem, some elementary algebraic geometryMathworldPlanetmathPlanetmath, and the fundamental theorem of projective geometryMathworldPlanetmath.


  • BPW P. Botta, S. Pierce, W. Watkins, Linear transformations that preserve the nilpotent matrices, Pacific J. Math. 104 (No. 1): 39–46 (1983).
Title Botta - Pierce - Watkins theorem
Canonical name BottaPierceWatkinsTheorem
Date of creation 2013-03-22 19:20:21
Last modified on 2013-03-22 19:20:21
Owner kammerer (26336)
Last modified by kammerer (26336)
Numerical id 6
Author kammerer (26336)
Entry type Theorem
Classification msc 15A04
Related topic FundamentalTheoremOfProjectiveGeometry
Related topic GerstenhaberSerezhkinTheorem