Spectral theory of copositive matrices

Let A ∈ Rn×n. We provide a block characterization of copositive matrices, with the assumption that one of the principal blocks is positive definite. Haynsworth and Hoffman showed that if r is the largest eigenvalue of a copositive matrix then r ⩾ ∣λ∣, for all other eigenvalues λ of A. We continue their study of the spectral theory of copositive matrices and show that a copositive matrix must have a positive vector in the subspace spanned by the eigenvectors corresponding to the nonnegative eigenvalues. Moreover, if a symmetric matrix has a positive vector in the subspace spanned by the eigenvectors corresponding to its nonnegative eigenvalues, then it is possible to increase the nonnegative eigenvalues to form a copositive matrix A′, without changing the eigenvectors. We also show that if a copositive matrix has just one positive eigenvalue, and n − 1 nonpositive eigenvalues then A has a nonnegative eigenvector corresponding to a nonnegative eigenvalue.