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Institute
In this thesis, the recognition problem and the properties of eigenvalues and eigenvectors of matrices which are strictly sign-regular of a given order, i.e., matrices whose minors of a given order have the same strict sign, are considered. The results are extended to matrices which are sign-regular of a given order, i.e., matrices whose minors of a given order have the same sign or are allowed to vanish. As a generalization, a new type of matrices called oscillatory of a specific order, are introduced. Furthermore, the properties for this type are investigated. Also, same applications to dynamic systems are given.
This thesis considers bounding functions for multivariate polynomials and rational functions over boxes and simplices. It also considers the synthesis of polynomial Lyapunov functions for obtaining the stability of control systems. Bounding the range of functions is an important issue in many areas of mathematics and its applications like global optimization, computer aided geometric design, robust control etc.
The Hadamard product of two matrices of the same order is obtained by entry-wise multiplication of their coefficients. In a similar way, the Hadamard power of a matrix and a polynomial is formed by real powers of their coefficients. Results for the Hadamard product of some important classes of matrices, e.g., positive definite matrices, conditionally negative definite matrices, and matrices with one positive eigenvalue are presented. The results are extended to give sufficient conditions for symmetric matrices to have exactly one positive eigenvalue. A Hurwitz (or stable) polynomial is a real polynomial whose roots are located in the open left half of the complex plane. Results for the Hadamard square root of Hurwitz polynomials of degree five are given. Also, a type of Oppenheim's inequality for Hurwitz matrices is presented. Finally, interval matrices, i.e., matrices with intervals as entries are studied, and new results for the interval property of several classes of matrices, e.g., inverse M-matrices, conditionally positive (negative) semidefinite matrices, and infinitely divisible matrices are given.
