A note on generalized invariant cones and the Kronecker canonical form (Q802701)

There are well-known conditions on the spectrum of a real \(n\times n\) matrix A which are equivalent to the existence of a proper cone \(K\subset {\mathbb{R}}^ n\) such that A(K)\(\subset K\). The authors generalize these results to the case of a pencil A-\(\lambda\) B, A,B \(m\times n\)-matrices. A cone \(K\subset {\mathbb{R}}^ n\) is A,B invariant if A(K)\(\subset B(K)\). Then the following statements are equivalent for A-\(\lambda\) B: (a) There exists a proper cone K s.t. A(K)\(\subset B(K).\) (b) The left Kronecker indices are zero, there are no infinite eigenvalues, \(\rho\) (J) \((=\) maximum modulus of the finite eigenvalues) is in the spectrum and has maximal index among all eigenvalues of modulus \(\rho\) (J).