It is proved that the set of all symmetric real matrices of order n with eigenvalues lying in the interval(α, β), denoted by Sn(α,β), is convex in Rn×n. With this result, some known results on positive(negati...It is proved that the set of all symmetric real matrices of order n with eigenvalues lying in the interval(α, β), denoted by Sn(α,β), is convex in Rn×n. With this result, some known results on positive(negative) definiteness, and Hurwitz(Shur) stability, as well as the aperiodic property of polytopes of symmetric matrices are generalized, and a series of insightful necessary and sufficient conditions for some general set of symmetric matrices contained in Sn(α,β) are presented,which are directly available for analysis of the positive(negative) definiteness, Hurwitz(Shur) stability and the aperiodic property of a wide class of sets of symmetric matrices.展开更多
文摘It is proved that the set of all symmetric real matrices of order n with eigenvalues lying in the interval(α, β), denoted by Sn(α,β), is convex in Rn×n. With this result, some known results on positive(negative) definiteness, and Hurwitz(Shur) stability, as well as the aperiodic property of polytopes of symmetric matrices are generalized, and a series of insightful necessary and sufficient conditions for some general set of symmetric matrices contained in Sn(α,β) are presented,which are directly available for analysis of the positive(negative) definiteness, Hurwitz(Shur) stability and the aperiodic property of a wide class of sets of symmetric matrices.
