84 research outputs found
Commutators from a hyperplane of matrices
Denote by the algebra of by matrices with entries in the
field . A theorem of Albert and Muckenhoupt states that every trace zero
matrix of can be expressed as for some pair of
matrices of . Assuming that and that has more than 3
elements, we prove that the matrices and can be required to belong to
an arbitrary given hyperplane of .Comment: 20 page
On Gerstenhaber's theorem for spaces of nilpotent matrices over a skew field
Let K be a skew field, and K_0 be a subfield of the central subfield of K
such that K has finite dimension q over K_0. Let V be a K_0-linear subspace of
n by n nilpotent matrices with entries in K. We show that the dimension of V is
bounded above by q n(n-1)/2, and that equality occurs if and only if V is
similar to the space of all n by n strictly upper-triangular matrices over K.
This generalizes famous theorems of Gerstenhaber and Serezhkin, which cover the
special case K=K_0.Comment: 17 pages (final version as will appear in Linear Algebra and its
Applications
Primitive spaces of matrices with upper rank two over the field with two elements
For fields with more than elements, the classification of the vector
spaces of matrices with rank at most is already known. In this work, we
complete that classification for the field . We apply the results
to obtain the classification of triples of locally linearly dependent operators
over , the classification of the -dimensional subspaces of
in which no matrix has a non-zero eigenvalue, and
the classification of the -dimensional affine spaces that are included in
the general linear group .Comment: 43 page
Large affine spaces of matrices with rank bounded below
Let K be an arbitrary (commutative) field with at least three elements, and
let n, p and r be positive integers with r<=min(n,p). In a recent work, we have
proved that an affine subspace of M_{n,p}(K) containing only matrices of rank
greater than or equal to r must have a codimension greater than or equal to
(r+1)r/2. Here, we classify, up to equivalence, these subspaces with the
minimal codimension (r+1)r/2. This uses our recent classification of the affine
subspaces of M_r(K) contained in GL_r(K) and which have the maximal dimension
r(r-1)/2.Comment: 27 pages (from v2: a minor error has been corrected, the proofs are
more detailed
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