Linear preservers of rank permutability
Let \(M_{n}(F)\) be the space of all \(n\times n\) matrices over a field \(F\). A \(k\)-tuple of matrices from \(M_{n}(F)\) is called ``rank permutable'' if the rank of the product of these \(k\) matrices is independent of the order of the factors (the concept arises in the problem of simultaneous triangualization of matrix families). An \(F\)-linear transformation \(T:M_{n}(F)\rightarrow M_{n}(F)\) is said to preserve rank permutability for \(k\)-tuples if, whenever a \(k\)-tuple \((A_{1},\dots,A_{k})\) is rank permutable, then \((T(A_{1}),\dots,T(A_{k}))\) is also rank permutable; it preserves rank permutablity strongly if the reverse implication is also true. The main theorem of the paper is that the following are equivalent: (1) for each \(k\), \(T\) preserves rank permutability strongly for \(k\)-tuples; (2) \(T\) is bijective and, for each \(k\), \(T\) preserves rank permutability for \(k\)-tuples; and (3) there exists an invertible matrix \(P\) and a nonzero scalar \(\alpha\) such that \(T(X)=\alpha PXP^{-1}\)for all \(X\) or \(T(X)=\alpha PX^{t}P^{-1}\) for all \(X\) (where \(X^{t}\) denotes the transpose).
scientific article; zbMATH DE number 2083495