Linear Preservers of Perimeters of Nonnegative Real Matrices
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For a nonnegative real matrix A of rank 1, A can be factored as ab^t for some vectors a and b. The perimeter of A is the number of nonzero entries in both a and b. If B is a matrix of rank k, then B is the sum of k matrices of rank 1. The perimeter of B is the minimum of the sums of perimeters of k matrices of rank 1, where the minimum is taken over all possible rank-1 decompositions of B. In this paper, we obtain characterizations of the linear operators which preserve perimeters 2 and k for some k≥4. That is, a linear operator T preserves perimeters 2 and k(≥4) if and only if it has the form T(A) = UAV , or T(A) = UA^tV with some invertible matrices U and V .
Abstract
For a nonnegative real matrix A of rank 1, A can be factored as ab^t for some vectors a and b. The perimeter of A is the number of nonzero entries in both a and b. If B is a matrix of rank k, then B is the sum of k matrices of rank 1. The perimeter of B is the minimum of the sums of perimeters of k matrices of rank 1, where the minimum is taken over all possible rank-1 decompositions of B. In this paper, we obtain characterizations of the linear operators which preserve perimeters 2 and k for some k≥4. That is, a linear operator T preserves perimeters 2 and k(≥4) if and only if it has the form T(A) = UAV , or T(A) = UA^tV with some invertible matrices U and V .
참고문헌 (8)
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