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SPACES OF MATRICES WITH SEVERAL ZERO EIGENVALUES
 

Summary: SPACES OF MATRICES WITH SEVERAL ZERO
EIGENVALUES
M. D. ATKINSON
Let V be an w-dimensional vector space over some field F, \F\ ^ n, and let SC be a
space of linear mappings from V into itself {SC ^ Horn (V, V)) with the property that
every mapping has at least r zero eigenvalues. If r = 0 this condition is vacuous but if
r = 1 it states that SC is a space of singular mappings; in this case Flanders [1] has
shown that dim#"^n(n -- 1) and that if dim #" = n(n -- 1) then either
SC = Hom(K, U) for some (n -- l)-dimensional subspace U of V or there exists
veV,v ^0 such that SC - {X e Horn {V, V): vX = 0}. At the other extreme r = n,
the case of nilpotent mappings, a theorem of Gerstenhaber [2] states that
dim SC ^ |n(n -1) and that if dim SC = $n(n -1) then SC is the full algebra of strictly
lower triangular matrices (with respect to some suitable basis of V).
In this paper we intend to derive similar results for a general value of r thereby
providing some common ground for the above theorems. The condition on the cardinal
of F plays an important role in our proofs although we do not know whether it is
essential to the theorems. Flanders required also the constraint char F ^ 2 at one point
in his proof; we do not require this and in fact our type of argument allows thefieldin
Flanders' theorem to have arbitrary characteristic.
THEOREM 1. Let V be an n-dimensional vector space over somefieldF, \F\ ^ n, and

  

Source: Atkinson, Mike - Department of Computer Science, University of Otago

 

Collections: Computer Technologies and Information Sciences