Normal forms of generic triangular band matrices and Jordan forms of nilpotent completions
Let \(F\) be an infinite field. For fixed integers \(k\) and \(n\), where \(1<k\leq n\), consider the set of triangular band matrices \({\mathcal D}_{k,n}\) formed by all the \(n\times n\) matrices \(A=[a_{ij}]_{i,j=1}^n\in F^{n\times n}\) satisfying the conditions: (1) \(a_{ij}=0\) for \(i>j\), (2) \(a_{ij}=0\) for \(j-i\geq k\), (3) \(a_{i,i+k-1}\not= 0\) for \(i=1,2,\dots ,n-k+1\). The authors study the possible Jordan forms of the nilpotent \(n\times n\) matrices with a given upper triangular part \(A\in {\mathcal D}_{k,n}\) with zero trace, that is, the nilpotent matrices of the form \(A+T\), where \(T=[t_{ij}]_{i,j=1}^n\in F^{n\times n}\) is strictly lower triangular (\(t_{ij}=0\) for \(i\leq j\)). For generic matrices within \({\mathcal D}_{k,n}\), the set of Jordan forms contains a unique minimal Jordan form. Moreover, in the generic case, there exists a nilpotent matrix in the orbit of a given upper triangular matrix having the minimal Jordan form, and also having the minimal rank among all matrices with the given upper triangular part. The orbit is understood in the sense of the triangular group action.
scientific article; zbMATH DE number 1443395