Linear preservers of copositive matrices

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.


Introduction
Matrix A ∈ M n (R) is called copositive if A T = A and x T Ax ≥ 0 whenever x ∈ R n , and x ≥ 0. If x T Ax > 0 whenever x ≥ 0 (entrywise), x = 0, then A is called strictly copositive. We write C (C n ) and SC (SC n ) to denote the two classes (with the subscript to indicate the size of the matrices, if useful). Of course C generalizes the positive semidefinite (PSD) and SC the positive definite (PD) matrices. Much is known about the copositive matrices [1][2][3] and there are tests for copositivity [2,4,5], but, in general, recognition is more difficult than for the PD/PSD cases and other related ones [6,7].
Let S n = S n (R) be the n(n + 1)/2-dimensional subspace of M n (R) consisting of symmetric matrices. In considering the action of linear transformations on copositive matrices, it suffices to consider linear transformations L: S n → S n (and it can be convenient because there are fewer variables to consider). However, it may also be convenient to consider L to be a linear map on M n (R). We shall do so interchangeably. We say that such a linear transformation preserves copositivity if A ∈ C implies L(A) ∈ C, and similarly for strict copositivity. More precisely, such an L is an into copositivity preserver. If L(C) = C, we have an onto copositivity preserver. Our purpose here is to better understand both types of linear copositivity preservers. The into preservers of the PSD matrices are fully understood, and they are recognized to be a difficult problem. The onto linear preservers of PSD are straight forwardly known to be the congruences by a fixed invertible matrix [8].
Certain natural kinds of linear transformations are more amenable to preserver analysis. We say a linear transformation on M n (R) is of standard form if there are fixed matrices R, S ∈ M n (R) such that L(A) = RAS (or L(A) = RA T S). Such a linear transformation is invertible if and only if R and S are invertible matrices. More generally, L is a linear transformation on M n (R) if L(A) = (l ij (A)), in which l ij is a linear functional in the entries of A. It is known that an invertible linear transformation that preserves rank is of standard form [9,10] and there are useful variations upon this sufficient condition.
Both the 'onto' and especially the 'into' copositivity linear preserver problems appear subtle. For example, in [11], a conjecture of N. Johnston is relayed: Any (into) copositivity preserver is of the form in which A i ≥ 0. Since any such map is (clearly) a copositivity preserver, this is a natural (though optimistic) conjecture. However, it is false even for 2-by-2 matrices. Suppose that a linear map on S 2 is given by If the argument is copositive, then a + b + 2c ≥ 0 (since it is the value of the quadratic form of the argument at 1 1 ), so that the image is nonnegative and, thus, copositive. But, as the conjectured form is a PSD preserver (by virtue of being a sum of congruences), the fact that 10 −1 −1 10 is PSD, while its image 10 18 18 10 is not, shows that our linear map is not of the conjectured form, though a copositivity preserver.

Some linear copositivity preservers
Here we indicate several types of linear copositivity preservers. Besides these, we do not know any others.

Linear copositivity preservers of standard form
Since our arguments are always in S n , a linear transformation of standard form is of the form Here we characterize both the into and onto preservers of copositivity of standard form. Such a transformation is invertible on S n if and only if R is invertible. First, a useful lemma about copositive matrices. We say that a vector v ∈ R n , n ≥ 2, is of mixed sign if v has both positive and negative entries. Nothing is assumed about 0 entries.

Lemma 2.1:
For each vector v ∈ R n of mixed sign, there is a matrix A ∈ C n such that v T Av < 0.
Proof: By the permutation similarity invariance of C n , we may assume that so that v T Av = 2v 1 v 2 < 0, as claimed.

Theorem 2.2:
Suppose that L is an into linear preserver of C n of standard form. Then, with S ∈ M n (R) and S ≥ 0.
Proof: Since L preserves symmetry and is of standard form, is not in C n . Hence, L is not a copositivity preserver. Thus, for any x ≥ 0, Sx must be weakly uniformly signed. This means that S ≥ 0 or ≤ 0. Since S appears twice, we may take it to be the former.
Since C n contains a basis of S n , a linear map on S n that is an onto copositivity preserver must be an invertible linear map and the inverse map must also be an onto preserver. If the map is of standard form, the inverse map just corresponds to inverting the S and S T (which, of course, must be invertible). Thus, we have that S −1 ≥ 0, as well as S ≥ 0 (or S −1 ≤ 0 and S ≤ 0). It is well known that this happens if and only if S is a monomial matrix, the product of a permutation matrix and a positive diagonal matrix. Taken together, this gives the following characterization of linear transformations of standard form that map C n onto itself.

Theorem 2.3:
Suppose that L is an onto linear preserver of C n of standard form. Then, in which S ∈ M n (R) and S is monomial.
Since C n forms a cone, we note that (i) any sum of into C n preservers is again an into C n preserver, though a sum of ones of standard form may no longer be of standard form, and (ii) the sum of onto C n preservers need no longer be onto. Also, it follows from the proven forms that both into and onto copositivity preservers of standard form are also (into and onto, respectively) PSD preservers.

Hadamard multiplier C n preservers
Recall that the Hadamard, or entrywise, product of two matrices A = (a ij ) and B = (b ij ) of the same size is defined and denoted by A • B = (a ij b ij ). If we consider a fixed n-by-n matrix H, then a natural type of linear transformation on n-by-n matrices A is given by We may also ask for which H are such transformations (into) C n preservers.
An n-by-n matrix B is called completely positive (CP) if B = FF T with F n-by-k and F ≥ 0. Thus, for F = [f 1 , f 2 , . . . , f k ] partitioned by columns, Then, the CP matrices, which also form a cone, are special PSD matrices. It is known [12] that the CP matrices are the cone theoretic dual of C n , as Tr( Now, consider a linear transformation of the form (1) with H a CP matrix of the form so that a linear transformation of the form (1) is also a sum of into transformations of standard form. With the exception of H being a rank 1, positive, symmetric matrix, such a transformation will not be onto.

General linear maps on S n
Let L(A) = (l ij (A)) in which each entry l ij is a linear functional in the entries of A. Symmetry requires that the functionals l ij and l ji be the same. It is possible to design such maps that are copositivity preservers (and in a similar way, PSD preservers). We note that, with this machinery, it is possible to design into, not onto, but invertible C n preservers. Here is an example. For and, for A ∈ C 2 , L(A) ≥ 0 and L(A) is PSD. So L is a C 2 preserver. Since l 12 ≥ 0, it is only into. However, L is invertible, as We note that more elaborate maps may be designed, including the possibility of negative off-diagonal entries.

PSD preservers that are not copositivity preservers
Of course, a PSD preserver need not be a copositivity preserver, and we note here a famous example of a PSD preserver that is not a copositivity preserver. It is known, and easily checked, that any 3-by-3 (symmetric) PSD matrix is transformed to another PSD matrix. Of course, L is not a fixed congruence, nor onto. However, note that a copositive matrix with 0 diagonal and positive off-diagonal entries is transformed to a matrix that is not copositive.
In general, copositivity preservers need not be PSD preservers and PSD preservers need not be copositivity preservers. However, we conjecture that onto copositivity preservers are also onto PSD preservers.

Onto linear copositivity preservers
We first make a fundamental observation about onto preservers of C n . Theorem 3.1: Let L : S n → S n be a linear transformation that maps C n onto C n . Then L is invertible and L −1 maps C n onto C n .
Proof: Since the copositive matrices include the standard basis for the symmetric matrices, their span is all of S n , which means that the map is invertible. Since L(C n ) = C n , application of L −1 to both sides of the equality yields the desired statement. Now we may see that linear onto preservers of C n also preserve several related sets. For B ∈ C n , let Proof: (a) and (b) follow since the copositive matrices are the closure of the strictly copositive matrices and an invertible linear transformation is continuous. (c) follows since SC n is the interior of C n . Now we show (d). Let R ∈ R and A ∈ C. Then R − αA ∈ C for some α > 0. So, L(R − αA) ∈ C and L(R) − αL(A) ∈ C. Since L is onto, L(A) runs over C when A runs over C. So L(R) ∈ R and L(R) ⊆ R. Similarly, Since both a fixed permutation similarity (equivalently, congruence) or a fixed positive diagonal congruence is an onto copositivity preserver, we have that monomial congruence is an onto copositivity preserver. We

Conjecture:
The onto copositivity preservers are exactly the fixed monomial congruences.
In a moment, we prove this in the 2-by-2 case. However, a proof, without additional assumptions, appears subtle. We have already shown that the conjecture holds if (1) the map is of standard form (Section 2.1). However, each of the following alternative additional assumptions is sufficient: (2) the map is an (onto) PSD preserver; (3) the map is rank nonincreasing (in which case it is of standard form [14,15]); (4) in the language of the Introduction, each of the component maps l ij is a function of only one entry of A (i.e. it is a monomial map on vec S n ); (5) each of the component maps l ij must have nonnegative coefficients.
We now study the onto 2-by-2 copositivity preservers. A symmetric matrix A ∈ S 2 is copositive if and only if The matrix A is strictly copositive if all the inequalities are strict.
From now on, we assume that L : S 2 → S 2 is a linear transformation that maps C 2 onto C 2 and we use the following notation with α 11 > 0 and α 22 > 0. In order to get a contradiction, suppose that α has this latter form.

Corollary 3.6:
If the linear transformation L : S 2 → S 2 maps C 2 onto C 2 then there is a monomial matrix E ∈ M n (R) such that L(A) = EAE T for any A ∈ S 2 .