Matrices for whichA^∗andA^†commute

Abstract

The class of complex star-dagger matrices for which A* and A t commute, is investigated. Some fundamental properties are given and a canonical form is derived. The interaction of this class of matrices with other common classes of special matrices, which as group matrices. EP matrices, and idempotents, will be looked into.

When is the hermitian/skew-hermitian part of a matrix a potent matrix?

“…Notice that N (A) ⊥ = R(A * ). Let us also remark that T and Y in (2.1) can be chosen so that they satisfy a useful relation [16,Corollary 6], providing a powerful tool for characterization of various classes of matrices [9]. However, we choose T and Y under the condition that Y * Y is diagonal, as we state in the following remark (in this case, the nice relation from [16,Corollary 6] is not preserved).…”

“…The next lemma will be crucial in our characterizations of the {k + 1}-potency of the Hermitian and skewHermitian parts (cf. [16,Corollary 6]). …”

On the Moore-Penrose inverse in rings with involution

“…We recall the reader that a 2 R is called (i) bi-normal if aa a a D a aaa ; (ii) bi-EP if a 2 R and aa a a D a aaa ; (iii) l-quasi-normal if aa a D a aa; (iv) r-quasi-normal if aaa D aa a; (v) l-quasi-EP if a 2 R and aa a D a aa; (vi) r-quasi-EP if a 2 R and aaa D aa a [2,4,6]. …”

On the elements aa† and a†a in

“…In [1, p. 10215] it was given a list of several equivalent conditions (involving the orthogonal projectors F F † and F † F ) for F to has the group inverse. The proof given therein relies in rank matrix theory and a matrix decomposition given by Hartwig and Spindelböck [4]. However, as we shall see, many of these equivalences can be stated in a ring setting, and proved by algebraic reasonings.…”