Abstract
ABSTRACT An n-by-n real symmetric matrix is called copositive if its quadratic form is nonnegative on nonnegative vectors. Our interest is in identifying which linear transformations on symmetric matrices preserve copositivity either in the into or onto sense. We conjecture that in the onto case, the map must be congruence by a monomial matrix (a permutation times a positive diagonal matrix). This is proven under each of some additional natural assumptions. Also, the into preservers of standard type are characterized. A general characterization in the into case seems difficult, and examples are given. One of them provides a counterexample to a conjecture about the into preservers.
Highlights
Matrix A 2 Mn(R) is called copositive if AT = A and xT Ax 0 whenever x 2 Rn, and x 0
Much is known about the copositive matrices
It may be convenient to consider L to be a linear map on Mn(R): We shall do so interchangeably. We say that such a linear transformation preserves copositivity if A 2 C implies L(A) 2 C, and for strict copositivity
Summary
It may be convenient to consider L to be a linear map on Mn(R): We shall do so interchangeably We say that such a linear transformation preserves copositivity if A 2 C implies L(A) 2 C, and for strict copositivity. It is known that an invertible linear transformation that preserves rank is of standard form [7, 12] and there are useful variations upon this su¢ cient condition. Both the “onto” and especially the “into” copositivity linear preserver problems appear subtle. 10 18 18 10 is not, shows that our linear map is not of the conjectured form, though a copositivity preserver
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