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Clément de Seguins Pazzis

Publications and source records attributed to Clément de Seguins Pazzis.

At least 73 records · Page 4Linked to original sources
arXiv2013

The classification of large spaces of matrices with bounded rank

Given an arbitrary (commutative) field K, let V be a linear subspace of M_n(K) consisting of matrices of rank lesser than or equal to some r nr-r+1 and #K>r, then either all the matrices of V vanish on some common (n-r)-dimensional subspace of K^n, or it is true of the matrices of its transpose V^t. Following some arguments of our recent proof of the Flanders theorem for an arbitrary field, we show that this result holds for any field. We also show that the results of Atkinson and Lloyd on the case dim V=n\,r-r+1 are independent on the given field save for the special case n=3, r=2 and #K=2, where the same techniques help us classify all the exceptional cases up to equivalence. Similar theorems of Beasley for rectangular matrices are also successfully extended to an arbitrary field.

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arXiv2013

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.

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arXiv2013

Spaces of matrices with few eigenvalues

Let K be a (commutative) field with characteristic not 2, and V be a linear subspace of n by n matrices that have at most two eigenvalues in K (respectively, at most one non-zero eigenvalue in K). We prove that the dimension of V is less than or equal to n(n-1)/2 provided that n be greater than 2 (respectively, the dimension of V is less than or equal to n(n-1)/2). We also classify, up to similarity, the linear subspaces of n by n matrices in which every matrix has at most two eigenvalues (respectively, at most one non-zero eigenvalue) in an algebraic closure of K and which have the maximal dimension among such spaces.

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arXiv2012

Spaces of matrices with a sole eigenvalue

Let K be an arbitrary (commutative) field and L be an algebraic closure of it. Let V be a linear subspace of M_n(K), with n>2. We show that if every matrix of V has at most one eigenvalue in K, then dim V<=1+n(n-1)/2. If every matrix of V has a sole eigenvalue in L and dim V=1+n(n-1)/2, we show that V is similar to the space of all upper-triangular matrices with equal diagonal entries, except if n=3 and K has characteristic 3, or if n=4 and K has characteristic 2. In both of those special cases, we classify the exceptional solutions up to similarity.

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arXiv2012

On the diagonal of the matrices in a similarity class

Let A be an n by n matrix with entries in an arbitrary field, and c_1,...,c_n be scalars. We prove that if A is not a scalar multiple of the identity matrix, then the condition c_1+...+c_n=tr(A) is necessary and sufficient for A to be similar to a matrix with diagonal entries c_1,...,c_n.

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arXiv2012

On commuting matrices and exponentials

Let A and B be matrices of M_n(C). We show that if exp(A)^k exp(B)^l=exp(kA+lB) for all integers k and l, then AB=BA. We also show that if exp(A)^k exp(B)=exp(B)exp(A)^k=exp(kA+B)$ for every positive integer k, then the pair (A,B) has property L of Motzkin and Taussky. As a consequence, if G is a subgroup of (M_n(C),+) and M -> exp(M) is a homomorphism from G to (GL_n(C),x), then G consists of commuting matrices. If S is a subsemigroup of (M_n(C),+) and M -> exp(M) is a homomorphism from S to (GL_n(C),x), then the linear subspace Span(S) of M_n(C) has property L of Motzkin and Taussky.

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arXiv2011

Sums of two triangularizable quadratic matrices over an arbitrary field

Let K be an arbitrary field, and a,b,c,d be elements of K such that the polynomials t^2-at-b and t^2-ct-d are split in K[t]. Given a square matrix M with entries in K, we give necessary and sufficient conditions for the existence of two matrices A and B such that M=A+B, A^2=a A+bI_n and B^2=c B+dI_n. Prior to this paper, such conditions were known in the case b=d=0, a<>0 and c<>0, and also in the case a=b=c=d=0. Here, we complete the study, which essentially amounts to determining when a matrix is the sum of an idempotent and a square-zero matrix. This generalizes results of Wang to an arbitrary field, possibly of characteristic 2.

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arXiv2011

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.

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arXiv2011

Large affine spaces of non-singular matrices

Let K be an arbitrary (commutative) field with at least three elements. It was recently proven that an affine subspace of M_n(K) consisting only of non-singular matrices must have a dimension lesser than or equal to n(n-1)/2. Here, we classify, up to equivalence, the subspaces whose dimension equals n(n-1)/2. This is done by classifying, up to similarity, all the n(n-1)/2-dimensional linear subspaces of M_n(K) consisting of matrices with no non-zero invariant vector, reinforcing a classical theorem of Gerstenhaber. Both classifications only involve the quadratic structure of the field K.

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arXiv2011

When does a linear map belong to at least one orthogonal or symplectic group?

Given an endomorphism u of a finite-dimensional vector space over an arbitrary field K, we give necessary and sufficient conditions for the existence of a regular quadratic form (resp. a symplectic form) for which u is orthogonal (resp. symplectic). A solution to this problem being already known in the case char(K)<>2, our main contribution lies in the case char(K)=2. When char(K)=2, we also give necessary and sufficient conditions for the existence of a regular symmetric bilinear form for which u is orthogonal. In the case K is finite with characteristic 2, we give necessary and sufficient conditions for the existence of an hyperbolic quadratic form (resp. a regular non-hyperbolic quadratic form, resp. a regular non-alternate symmetric bilinear form) for which u is orthogonal.

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arXiv2011

The linear preservers of non-singularity in a large space of matrices

Let K be an arbitrary (commutative) field, and V be a linear subspace of M_n(K) such that codim V PMQ or M->PM^TQ for some pair (P,Q) of non-singular matrices of M_n(K), unless n=3, codim V=1 and K is isomorphic to F_2. This generalizes a classical theorem of Dieudonné with a similar strategy of proof. Weak linear preservers are also discussed, as well as the exceptional case of a hyperplane of M_3(F_2).

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arXiv2011

The linear preservers of real diagonalizable matrices

Using a recent result of Bogdanov and Guterman on the linear preservers of pairs of simultaneously diagonalizable matrices, we determine all the automorphisms of the vector space M_n(R) which stabilize the set of diagonalizable matrices. To do so, we investigate the structure of linear subspaces of diagonalizable matrices of M_n(R) with maximal dimension.

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arXiv2010

To what extent is a large space of matrices not closed under the product?

Let K denote a field. Given an arbitrary linear subspace V of M_n(K) of codimension lesser than n-1, a classical result states that V generates the K-algebra M_n(K). Here, we strengthen this in three ways: we show that M_n(K) is spanned by the products of the form AB with A and B in V; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear hyperplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.

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arXiv2010

On the matrices of given rank in a large subspace

Let V be a linear subspace of M_{n,p}(K) with codimension lesser than n, where K is an arbitrary field and n >=p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K is isomorphic to F_2. Here, we give a sufficient condition on codim V for V to be spanned by its rank r matrices for a given r between 1 and p-1. This involves a generalization of the Gerstenhaber theorem on linear subspaces of nilpotent matrices.

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arXiv2010

Equivariant K-theory for proper actions of non-compact Lie groups

Generalizing a construction of Wolfgang Lück and Bob Oliver, we define a good equivariant cohomology theory on the category of proper G-CW complexes when G is an arbitrary Lie group (possibly non-compact). This is done by constructing an appropriate classifying space that arises from a Gamma-G-space. It is proven that this theory effectively generalizes Segal's equivariant K-theory when G is compact.

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arXiv2010

A classifying space for Phillips' equivariant K-theory

In a previous paper, we have constructed, for an arbitrary Lie group G and any of the fields F=R or C, a good equivariant cohomology theory KF_G^*(-) on the category of proper $G$-CW-complex and have justified why it deserved the label ``equivariant K-theory". It was shown in particular how this theory was a logical extension of the construction of Lück and Oliver for discrete groups and coincided with Segal's classical K-theory when G is a compact group and only finite G-CW-complexes are considered. Here, we compare our new equivariant K-theory with that of N.C. Phillips: it is shown how a natural transformation from ours to his may be constructed which gives rises to an isomorphism when G is second-countable and only finite proper G-CW-complexes are considered. This solves the long-standing issue of the existence of a classifying space for Phillips' equivariant K-theory.

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arXiv2010

On small matrix subalgebras with a trivial centralizer

Given an integer n greater of equal to 3, we investigate the minimal dimension of a subalgebra of M_n(K) with a trivial centralizer. It is shown that this dimension is 5 when n is even and 4 when it is odd. In the latter case, we also determine all 4-dimensional subalgebras with a trivial centralizer.

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