A stratification of the set of normal matrices (Q2784351)

The set of \(N\) of normal matrices in \(\mathbb{R}^{n\times n}\) is considered as a stratified subset of \(\mathbb{R}^{2n}\). Every normal matrix \(Z\) has a Toeplitz decomposition \(Z= H+ iK\) where \(H\), \(K\) are Hermitian matrices. Based on this, the author constructs a stratification with strata of dimension \(n^2+ j\) for \(1\leq j\leq n\). The stratum of maximal dimension \(n^2+ n\) can be parameterized in the form \(Z= H+ ip(H)\) for a polynomial \(p\) with real coefficients. This is used to develop a new computational algorithms that involve normal matrices. In particular, eigenvalues of large, possibly sparse, normal matrices an be computed by a generalization of the Lanczos method for Hermitian matrices.NEWLINENEWLINENEWLINEThe proposed method is compared with the Arnoldi method and several extensive numerical tests are given.