The Block Numerical Range Of Matrix Polynomials

This thesis mainly deals with the numerical range of matrix polynomials.There has been many studies on the numerical range of matrix with rewardingresults, which are widely applied in areas of convergence analysis of iterativemethod and location and sensitivity analysis of eigenvalue. In recent years, stud-ies on the quadratic numerical range of 2×2 block operator matrix and blockrange of n×n block operator matrix in Hilbert spaces are obtained. In the fieldof the numerical range of matrix polynomials, Gohberg and his cooperators haveproposed the concept of the numerical range of matrix polynomials; Li and hiscooperators have carried out a systematic research on the numerical range of ma-trix polynomials. However, results on the quadratic numerical range of matrixpolynomials are rarely seen. This thesis expands the concept of the block numer-ical range of matrix to situations of matrix polynomials, and provides relevanttheoretic analysis. It is hoped that the results will be helpful to the research incontrol theory and eigenvalue analysis. This thesis consists of five chapters.Chapter 1 serves as a general review of the background and developmentof the numerical range of matrix and matrix polynomials, as well as a briefintroduction to the contents of this thesis.Chapter 2 mainly deals with the quadratic numerical range of matrix poly-nomials. The concept of the quadratic numerical range of matrix polynomialsis defined in this part. The inclusive relationship of the eigenvalue of matrixpolynomials, the numerical range of matrix polynomials and the quadratic nu-merical range of matrix polynomials is concluded while the su?ciency conditionof the boundness of the quadratic numerical range of matrix polynomials is given.Lastly, the geometry characteristics of the quadratic numerical range of matrixpolynomials are studied.Chapter 3 defines the n×n block numerical range of matrix polynomialsand provides the inclusive relationship of the eigenvalue, the numerical range and the block numerical range, reaching the conclusion that further block partitionof matrix polynomials can reduce the block numerical range.Chapter 4 is a further research of the bound of the eigenvalues of matrixpolynomials. A uniform way of calculating the super bound of the eigenvalueof matrix polynomials is given. Estimation of the bound of the eigenvalue isobtained through a reasonable partition of the companion matrix of the matrixpolynomials. Finally, relationship between the block numerical range and certaineigenvalue of companion matrix of the matrix polynomials is presented.Chapter 5 includes the numerical examples, the results of which show theinclusive relationship between the numerical range of matrix polynomials and thequadratic numerical range of matrix polynomials. The geometry characteristicsare also testified by the examples. Reasonable partition can result in smallernumerical ranges that include the eigenvalue.