Orthogonal polyanalytic polynomials and normal matrices
Mathematics of Computation
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In this paper we consider the set of normal matrices ${\cal N} \subset \Bbb C^{n \times n}$ as a stratified submanifold of $\Bbb R^{2n^2}$. Based on the Toeplitz decomposition, we construct a stratification of ${\cal N}$ with the strata of dimension $n^2+j$ for $1 \leq j \leq n$. The stratum of the maximal dimension n2+n is readily parametrizable since the Toeplitz decomposition Z=H+iK of a generic $Z \in {\cal N}$ equals Z=H+ip(H) for a polynomial p with real coefficients. Using this, it is possible to approach computational tasks involving normal matrices in a new way. To give an example, we consider the problem of approximating eigenvalues of a large, possibly sparse, normal matrix Z. In particular, we generalize the Hermitian Lanczos method to normal matrices.