Linear operators leaving a class of matrices with fixed singular values invariant

Let , and let be the set F m×n of all m × n matrices over F. Given nonzero D ∈ M, denote by S(D) the collection of all matrices in having the same singular values as D. We show that if φ is a nonsingular linear operator on that satisfies φ(S(D)) ⊂ S(D), then φ(S(D)) and, with a previous result, the structure of φ is determined. Moreover, it is shown that the nonsingularity assumption can be removed except for the case when with either rank(D) = 1 or all the singular values of D are equal. In the exceptional cases, examples of singular linear operators φ satisfying are given. We also study those linear operators on satisfying φ(S(D)) ⊂ S(C) for some C,D ∈ .