Study On The Numerical Characteristics Of Matrices And Judgment For Some Special Matrices

Matrix theory has broad applications in modern mathematics, physics,management science and engineering, economics, biology, auto-control system, imageanalysis etc. In this thesis, the estimation for bounds of some numerical characteristicsof the matrix which contain the sum of the squares of the module of eigenvalues、rank、eigenvalues、spread and simple judgment for stable matrix, positive-definite matrix andM-matrix are researched. The main contents and innovations are as follows:1. The complex matrices are partitioned as the following formwhereAk×kis k th leading principal submatrix of M,1≤k≤n1,x≠0. We educe theestimation for the sum of the squares of the module of eigenvalues Furthermore,based on this formula, the upper bounds for the module of the determinant of matrix areobtained.2. The lower bounds for the rank of matrix are researched, and the new estimationsof lower bounds for the rank are obtained. Some sufficient conditions for nonsingularmatrix are given.3. For arbitrarily complex matrix M∈Cn×n, we prove that all eigenvalues of M areonly included in the following diskThe estimations for spread of matrix are proposed.4. Based on the Schur theorem and inequality theory, we get several estimations forthe real and imaginary parts of eigenvalues of matrix and improve some previousresults.5. We define a new sign-preserving transformation (called “sequence-sign-preserving transformation”) mainly on the basis of the elementary transformation of the matrix, deducing the simple judgment for stable matrix, positive-definite matrix andM-matrix. Moreover, some sufficient conditions for judgment for stable matrix,positive-definite matrix and M-matrix are given.