The number of Kronecker indices of square pencils of a special kind
Let \({\mathcal A},{\mathcal B}\) be similarity classes of \(n\times n\) matrices over a field \(F\). The authors prove that the ranks of the matrix pencils \(xA+B\) \((A\in {\mathcal A},B\in {\mathcal B})\) form the closed interval with end points \(\max\{r_{\mathcal A},r_{\mathcal B}\}\) and \(\min\{r_{\mathcal A}+r_{\mathcal B},n\}\) where \(r_{\mathcal A},r_{\mathcal B}\) are the ranks of the members of \({\mathcal A},{\mathcal B}\). Since the number of Kronecker indices of \(xA+B\) is its nullity \(n-\text{rank}(xA+B)\), this result is of some significance for the classification of the pencils in question.
scientific article; zbMATH DE number 1419710