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Remarks on the sharp partial order and the ordering of squares of matrices. (English) Zbl 1103.15013
This paper characterizes certain partial orders on complex matrices and discusses their mutual relationships, as well.
Two complex $$n$$-square matrices $$A$$ and $$B$$ are said to satisfy the star partial order $$A \overset{*}{\leq} B$$ if $$B A^{\dagger} = A A^{\dagger}$$ and $$A^{\dagger} B = A^{\dagger} A$$ hold, where $$A^{\dagger}$$ denotes the Moore-Penrose inverse of $$A.$$ This order was introduced by M. P. Drazin [Bull. Am. Math. Soc. 84, 139–141 (1978; Zbl 0395.20044)]. It is a well known fact that, in the definition above, $$A^{\dagger}$$ may be replaced by $$A^*,$$ the conjugate transpose of $$A.$$
Let $${\mathbb C}_n^{GP}$$ denote the set of all complex $$n$$-square matrices $$A$$ with a group inverse $$A^\#.$$ Among others, the author proves the following: Let $$A \in {\mathbb C}_n^{GP}$$ with $$\text{ rank}(A) < n,$$ and let $$B \in {\mathbb C}^{n \times \, n}.$$ Then $$A \overset{*}{\leq} B$$ and $$AB = BA$$ hold if and only if $$A$$ and $$B$$ are simultaneously unitarily similar to certain 2-square block matrices with same (1,1) and (2,1) blocks.
Two matrices $$A, B \in {\mathbb C}_n^{GP}$$ are said to satisfy the sharp partial order $$A \overset{\#}{\leq} B$$ if $$B A^\# = A A^\#$$ and $$A^\# B = A^\# A.$$ This order was introduced by S. K. Mitra [Lin. Alg. Appl. 92, 17–37 (1987; Zbl 0619.15006)]. It is shown that, if $$A, B \in {\mathbb C}_n^{GP},$$ any two of the following statements imply the third one: (i) $$A\overset{*}{\leq} B.$$ (ii) $$A^2 \overset{*}{\leq} B^2.$$ (iii) $$A \overset{\#}{\leq} B.$$
Other investigations concern relationships between the star and the sharp partial orders (including the case of coincidence), relationships of these orders with the so-called minus partial order, and consequences for spectral properties of the underlying matrices.

##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 15B57 Hermitian, skew-Hermitian, and related matrices 15A09 Theory of matrix inversion and generalized inverses 15A06 Linear equations (linear algebraic aspects)
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##### References:
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