Page 144 - Textos de Matemática Vol. 44

134 T. MORAN AND I. SPITKOVSKY
3. Almost normality of A∗
The definition of normality immediately implies that a matrix A is normal if and only simultaneously with A∗. We brought up this trivial observation to emphasize that for almost normality the situation changes.
Theorem 3.1. Let A be an almost normal matrix. Then A∗ is almost normal if and only if its PAN-rank is at most 2.
Proof. Using the decomposition (2.1) for A, we observe that A∗ is unitarily similar to A∗n ⊕A∗a. Since A∗n is normal along with An, A∗ will be almost normal only simultaneously with A∗a.
If the PAN-rank of A is 2, then A∗a is almost normal, as any 2-by-2 matrix. If the PAN-rank is zero, the situation is even simpler: A∗ is normal, since A is. This proves the sufficiency.
Suppose now that the PAN-rank of A is at least 3. Using the canonical form (1.1) of Aa, we see that A∗a is unitarily similar to
λ1 0 ... 0 0 0 λ2 ... 0 0
 . . (3.1)  . ..
0 0 ... λn−1 0 β1 β2 ... βn−1 μ
The eigenvalues λj of the matrix (3.1) have geometric multiplicities one, and the respective eigenvectors, up to scalar multiples, are
(3.2) xj =[0 ... 0λj −μ0 ...0βj]T
(λj −μ in the jth position), j = 1,...,n−1. If μ is different from all λj, there is also an eigenvector xn = en corresponding to the eigenvalue μ of (3.1); otherwise the set {x1 , . . . , xn−1 } of the eigenvectors of (3.1) is complete.
Either way, no pair of the eigenvectors xi,xj is orthogonal. Thus, starting with n = 3, the matrix (3.1) is not almost normal. This proves the necessity.
4. Self-commutator
Yet another way to define normality is as follows: a matrix A is normal if and only if its self-commutator [A] has rank zero. Since [A] is traceless for any A, non-normal matrices have self-commutators of rank at least two. On the other hand, it was shown in [6, Section 4] that for almost normal matrices the rank of self-commutators does not exceed three. Here is an ε-improvement of this result.