Extremal sparsity of the companion matrix of a polynomial
Let \(C\) be the companion matrix of a monic polynomial \(p\) over a field \(\mathbb F\). The authors prove that if \(A\) is a matrix whose entries are rational functions of the coefficients of \(p\) over \(\mathbb F\) and whose characteristic polynomial is \(p\), then \(A\) has at least as many nonzero entries as \(C\). In other words, the companion matrix is the sparsest matrix of the mentioned set of matrices \(A\).
scientific article; zbMATH DE number 6127366