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For a rank- matrix , we define the perimeter of as the number of nonzero entries in both and . We characterize the linear operators which preserve the rank and perimeter of rank- matrices over semifields. That is, a linear operator preserves the rank and perimeter of rank- matrices over semifields if and only if it has the form , or with some invertible matrices U and V.
We investigate the perimeter of nonnegative integer matrices. We also characterize the linear operators which preserve the rank and perimeter of nonnegative integer matrices. That is, a linear operator preserves the rank and perimeter of rank- matrices if and only if it has the form , or with appropriate permutation matrices and and positive integer matrix , where denotes Hadamard product.
Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of by . This includes the usual product and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying such that ϕ: V → V has the form
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