Criterion For H Matrix And The Distribution Of Eigenvalues

With the rapid development of science and technology, the matrix theoryis being used more and more extensively in many disciplines such as computa-tional mathematics, system engineering and control theory. Since it can revealthe essence profoundly and express in a simple way, solving the complex engi-neering problem with matrix theory and methods has attracted more and moreattention in the area of engineering. The research on matrix theory and appliancehas become one of the hot topics in science and engineering.H matrix is a important class of special matrix, they have many special prop-erties and play an important role in many areas such as computational mathe-matics, numerical linear algebra and control theory. In the area of studying thedistribution and estimation of eigenvalues with the special properties of H matrix,many domestic and overseas scholars have done a great of works and obtainedlots of important results.In this paper, we first study the properties of H matrix, presenting some newcriteria for H matrix by constructing positively diagonal matrix factors progres-sively, which improve and extend some relative results. Meanwhile, we give somenew bounds for the eigenvalues of block matrix by using the properties of blockdiagonally dominant matrix and G-function. Furthermore, through constructingspecial matrix, using the Gers?gorin disc theorem and the Ostrowski disc the-orem, we obtain some bounds for the eigenvalues of a special class of matrix'sSchur complements, which are expressed by the original matrix's elements.In chapter one, we first present some background knowledge and recent worksfor eigenvalues and H matrix. Then we introduce some basic symbols and defini-tions used in this paper.In chapter two, we obtain some new criteria for generalized strictly dominantmatrix by constructing positively diagonal matrix factors progressively and em-ploying inequalities techniques. And we extend these methods to the conditionsof irreducible matrix and non-zero chain matrix. Thus we improve some existingresults effectively.In chapter three, we first consummate some recent results about the distri-bution of the eigenvalues for block matrix by combining the properties of blockdiagonally dominant matrix and G-function. Then we present some new boundsfor eigenvalues of block matrices by taking advantage of some properties of block diagonally dominant matrix. The superiority of our new results is demonstratedby some numeric examples.In chapter four, through constructing special matrix, by employing inequali-ties techniques, the properties of Schur complements, the Gers(?)gorin disc theoremand the Ostrowski disc theorem, we obtain some bounds for the eigenvalues of aspecial class of matrix's Schur complements, which are expressed by the originalmatrix's elements.