Given a square matrix A, an A-invariant subspace is called hyperinvariant (respectively, characteristic) if and only if it is also invariant for all matrices T (respectively, nonsingular matrices T) that commute with A. Shodaʼs Theorem gives a necessary and sufficient condition for the existence of characteristic non-hyperinvariant subspaces for a nilpotent matrix in GF(2). Here we present an explicit construction for all subspaces of this type.Peer ReviewedPostprint (published version

We prove that the local A 1 -degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local A 1 -degree over the residue field. This fact was originally suggested by Morel's work on motivic transfers, and by Kass and Wickelgren's work on the Scheja-Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren relating the Scheja-Storch form and the local A 1 -degree.

Let F be an arbitrary field, Mn(F) the set of all matrices n×n over F and J∈Mn(F) a Jordan matrix. In this paper, we obtain an explicit formula for the determinant of any matrix that commutes with J, i.e., the determinant of any element T∈Z(J), the centralizer of J. Our result can also be extended to any T′∈Z(A), where A∈Mn(F), can be reduced to J=S−1AS. This is because T=S−1T′S∈Z(J), and clearly View the MathML source. If F is algebraically closed, any matrix A can be reduced in this way to a suitable J.\ud \ud In order to achieve our main result, we use an alternative canonical form W∈Mn(F) called the Weyr canonical form. This canonical form has the advantage that all matrices K∈Z(W) are upper block triangular. The permutation similarity of T∈Z(J) and K∈Z(W) is exploited to obtain a formula for the determinant of T.\ud \ud The paper is organized as follows: Section 2 contains some definitions and notations that will be used through all the paper. In Section 3, matrices T∈Z(J) are described and the determinant of T is computed in a particular case. In Section 4, we recall the Weyr canonical form W of a matrix and the corresponding centralizer Z(W). A formula to compute the determinant of any K∈Z(W) is rewritten. Finally, in Section 5 an explicit formula for the determinant of any T∈Z(J) is obtained.Peer ReviewedPostprint (published version

Given an endomorphism A over a finite dimensional vector space having Jordan-Chevalley decomposition, the lattices of invariant and hyperinvariant subspaces of A can be obtained from the nilpotent part of this decomposition. We extend this result for lattices of characteristic subspaces. We also obtain a generalization of Shoda's Theorem about the characterization of the existence of characteristic non hyperinvariant subspaces.

The main purpose of this paper is to study common invariant subspaces of any matrix in the centralizer of a given matrix A∈Mn(F), where F denotes an algebraically closed field. In particular, we obtain a necessary and sufficient condition for the existence of a common eigenvector for all the matrices in this set.Peer ReviewedPostprint (published version