Determinant of a matrix that commutes with a Jordan matrix
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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. 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. 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.
CitationMontoro, M.; Ferrer, J.; Mingueza, D. Determinant of a matrix that commutes with a Jordan matrix. "Linear algebra and its applications", 30 Octubre 2013, vol. 439, núm. 12, p. 3945-3954.