Page 141 - Textos de Matemática Vol. 44

ON ALMOST NORMAL MATRICES
TYLER MORAN AND ILYA SPITKOVSKY
For Nata´lia Bebiano on her 60th birthday
Abstract. Let an n-by-n matrix A be almost normal in a sense that it has n − 1 orthogonal eigenvectors. The properties of its numerical range W (A) and Aluthge transform ∆ are explored. In particular, it is proven that for unitarily irreducible almost normal A, W(A) cannot have flat portions on the boundary and ∆(A) is not normal (the latter, under the additional conditions that n > 2 and A is invertible). In passing, the unitary irreducibility criterion for A, almost normality criterion for A∗, and the rank formula for the self- commutator A∗A − AA∗ are established.
1. Introduction
First, let us fix the notation. Throughout this paper, Cn (resp., Cn×n) will represent the vector space of all n-vectors (resp., the algebra of all n-by-n matrices) with complex entries. We also let e1, . . . , en denote the standard basis of Cn.
A matrix A ∈ Cn×n is normal if it commutes with A∗. For a non-normal A, several measures of non-normality can be used to characterize its deviation from being normal. They include the distance from A to the set of all normal matrices, various norms of the self-commutator [A] := A∗A − AA∗, etc. (see, e.g., [3]). More recently, the following notion was introduced in [6]: A ∈ Cn×n is almost normal if it has (at least) n − 1 pairwise orthogonal eigenvectors. Equivalently, almost normal matrices are unitarily similar to
λ1 0 ... 0 β1  0 λ2 ... 0 β2 
. .  (1.1) . ..
 0 0 . . . λ n − 1 β n − 1  00...0μ
2010 Mathematics Subject Classification. Primary 15B57; Secondary 15A60
Key words and phrases. Almost normal matrix, numerical range, Aluthge transform This research was supported in part by the NSF CSUMS grant DMS–0703532.
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