Gerstenhaber - Serezhkin theorem

Let 𝔽 be an arbitrary field. Consider β„³n⁒(𝔽), the vector spaceMathworldPlanetmath of all nΓ—n matrices over 𝔽. Define

  • β€’

    𝒩={Aβˆˆβ„³n⁒(𝔽):A⁒is nilpotent},

  • β€’

    𝒒⁒ℒn⁒(𝔽)={Aβˆˆβ„³n⁒(𝔽):det⁑(A)β‰ 0},

  • β€’

    𝒯={Aβˆˆβ„³n⁒(𝔽):A⁒is strictly upper triangular}.

Notice that 𝒯 is a linear subspace of β„³n⁒(𝔽). Moreover, π’―βŠ†π’© and dim⁑𝒯=n⁒(n-1)/2.

The Gerstenhaber – Serezhkin theorem on linear subspaces contained in the nilpotent cone [G, S] reads as follows.

Theorem 1

Let L be a linear subspace of Mn⁒(F). Assume that LβŠ†N. Then
(i) dim⁑L≀n⁒(n-1)/2, (ii) dim⁑L=n⁒(n-1)/2 if and only if there exists U∈G⁒Ln⁒(F) such that {U⁒A⁒U-1:A∈L}=T.

An alternative simple proof of inequality (i) can be found in [M].


  • G M. Gerstenhaber, On nilalgebras and linear varieties of nilpotent matricesMathworldPlanetmath, I, Amer. J. Math. 80: 614–622 (1958).
  • M B. Mathes, M. Omladič, H. Radjavi, Linear Spaces of Nilpotent Matrices, Linear AlgebraMathworldPlanetmath Appl. 149: 215–225 (1991).
  • S V. N. Serezhkin, On linear transformations preserving nilpotency, VestsΞΉΒ― Akad. Navuk BSSR Ser. FΞΉΒ―z.-Mat. Navuk 1985, no. 6: 46–50 (Russian).
Title Gerstenhaber - Serezhkin theorem
Canonical name GerstenhaberSerezhkinTheorem
Date of creation 2013-03-22 19:20:05
Last modified on 2013-03-22 19:20:05
Owner kammerer (26336)
Last modified by kammerer (26336)
Numerical id 7
Author kammerer (26336)
Entry type Theorem
Classification msc 15A30
Related topic BottaPierceWatkinsTheorem