Page 142 - Textos de Matemática Vol. 44

132 T. MORAN AND I. SPITKOVSKY
Using an additional diagonal unitary similarity, it is possible to arrange that βj ≥ 0, j = 1,...,n − 1, while of course the eigenvalues λ1,...,λn−1,μ are defined by A uniquely.
In this paper, we study aspects of almost normality not covered by [6]. The first three sections are short and rather elementary. Namely, Section 2 provides the criterion of unitary irreducibility, Section 3 describes those almost normal matrices A for which A∗ is also almost normal, and Section 4 contains the formula for the rank of [A]. The last two sections are devoted to the numerical range (Section 5) and the Aluthge transform (Section 6) of almost normal matrices.
2. Pure almost normality
Using a unitary similarity to represent A as a direct sum of the maximal possible number of blocks, we see that every almost normal matrix A is unitarily similar to
(2.1) An ⊕Aa,
where the block An is normal while Aa is almost normal and unitarily irre- ducible. The blocks An and Aa in (2.1) are each defined up to unitary similar- ity.
If A itself is normal (which is not excluded by the formal definition of almost normality), the block Aa disappears from (2.1). In the other extreme, that is, if A = Aa, we will say that A is pure almost normal. We will refer to the size of the block Aa in (2.1) as the PAN-rank of A. Thus, an almost normal n-by-n matrix is normal (resp., pure almost normal) if its PAN-rank is 0 (resp, n). Note also that the PAN-rank cannot equal one.
In terms of representation (1.1), pure almost normal matrices can be char- acterized as follows.
Theorem 2.1. An almost normal matrix A is unitarily irreducible if and only if in its canonical form (1.1) all βj are different from zero and all λj are distinct:
(2.2) βj ̸=0, λi ̸=λj (i,j=1,...,n−1; i̸=j).
Proof. Necessity. If βm = 0 for some m, then A under a permutational simi- larity corresponding to the (1, m) transposition turns into the direct sum of a one dimensional block (λm) with an almost normal matrix from C(n−1)×(n−1) and thus is unitarily reducible. On the other hand, if λi = λj, then under an appropriate unitary similarity affecting only i, j-th rows and columns we obtain from (1.1) a matrix with the same first n − 1 columns and the zero (i, n)-entry. Thus, the already proven part of the statement applies.