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Monotone maps on diagonalizable matrices. (English) Zbl 1310.15047
Let \(\mathbb{F}\) be an algebraically closed field and let \(M_{n}(\mathbb{F})\) denote the set of all \(n\times n\) matrices with entries from \(\mathbb{F}\). The group inverse of \(A\in \) \(M_{n}(\mathbb{F})\), if it exists, is denoted by \(A^{\#}\). Recall that the set of matrices having a group inverse is precisely the set \(I_{n}^{1}(\mathbb{F})=\{A\in M_{n}(\mathbb{F} ):\mathrm{rank}(A)=\mathrm{rank}(A^{2})\}\) of index one matrices. For \(A,B\in M_{n}(\mathbb{F})\) we write \(A\leq ^{\#}B\) if \(A=B\) or \(A\in I_{n}^{1}(\mathbb{F})\) and \( AA^{\#}=BA^{\#}=A^{\#}B\). Moreover, if \(A\leq ^{\#}B\) and \(A\neq B\), then we write \(A<^{\#}B\). Let \(D_{n}(\mathbb{F})\) denote the set of all diagonalizable matrices in \(M_{n}(\mathbb{F})\).
In the paper, two (main) results are proved. The authors first characterize injective maps on \(D_{n}(\mathbb{F)}\), \(n\geq 3\), that are monotone with respect to the \(\leq ^{\#}\)-order, i.e. injective maps \( T:D_{n}(\mathbb{F)\rightarrow }D_{n}(\mathbb{F)}\), \(n\geq 3\), where \(A\leq^{\#}B\) implies \(T(A)\leq ^{\#}T(B)\), \(A,B\in D_{n}(\mathbb{F)}\). The proof of the first result is divided into several steps. Suppose \(T\) is of the above form. The authors first show that \(T\) preserves rank and that for any \( \lambda \in \mathbb{F}\) there exists \(\mu \in \mathbb{F}\) such that \(T(\lambda I)=\mu I\). Here, \(I\in M_{n}(\mathbb{F})\) is the identity matrix. The authors next show that \(T\) is \(0\)-additive and finally obtain the form of \(T\). Throughout the proof, they use spectrally orthogonal matrix decompositions. The second result of the paper describes the form of injective maps on \(D_{n}(\mathbb{F)}\), \(n\geq 3\), that are strongly monotone with respect to the \(<^{\#}\)-order, i.e. injective maps \(T:D_{n}(\mathbb{ F}\rightarrow D_{n}(\mathbb{F})\), \(n\geq 3\), where \(A<^{\#}B\) if and only if \(T(A)<^{\#}T(B)\), \(A,B\in D_{n}(\mathbb{F)}\).
The paper has four sections. The main results are presented in the first section while some preliminaries are provided in the second section. Proofs are given in the third section. The authors provide in the last section three examples with the aim to show that the assumptions of the two theorems are indispensable.
15A86 Linear preserver problems
15A04 Linear transformations, semilinear transformations
15A09 Theory of matrix inversion and generalized inverses
06A06 Partial orders, general
15B48 Positive matrices and their generalizations; cones of matrices
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