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The relationship among eigenvalues of a given square matrix A and the rank one updated matrix A + v k q ⁎ , where v k is an eigenvector of A associated with the eigenvalue λ k and q is an arbitrary vector, was described by Brauer in 1952. In this work we study the relations between the Jordan structures of A and A + v k q ⁎ . More precisely, we analyze the generalized eigenvectors of the updated matrix in terms of the generalized eigenvectors of A , as well as the Jordan chains of the updated matrix. Further, we obtain similar results when we use a generalized eigenvector of A instead of the eigenvector v k . [ABSTRACT FROM AUTHOR]

Given a square matrix A, a Brauer's theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75-91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt's and Hotelling's deflations. An extension of the aforementioned Brauer's result, Rado's theorem, shows how to modify r eigenvalues of A at the same time via a rank- r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem. [ABSTRACT FROM AUTHOR]

This paper deals with some properties of α1-matrices and α2-matrices which are subclasses of nonsingular H-matrices. In particular, new characterizations of these two subclasses are given, and then used for proving algebraic properties related to subdirect sums and Hadamard products. [ABSTRACT FROM AUTHOR]