Estimation On Eigenvalues Of The Hadamard Product Of Matrices And Perturbation Bounds For Characteristic Polynomials

The perturbation problem of the generalized eigenvalues is mainly to study theeigenvalues and eigenvectors how changes due to changes in the matrix elements,namely that is eigenvalue stability depends on the matrix elements, rather than relyingon algorithms. In numerical calculation，actually we get the data with errors，and enteredthem into the computer will generate errors, then the final results must be inaccurate. Inaddition, we know the characteristic polynomial coefficients of a complex matrix havean important role in quantum physics. Now we have many algorithms to calculate thecoefficients, in order to assess the stability of these algorithms, we need to know thecoefficientck error bounds and sensitivity to perturbations of the matrix. We willintroduce that in the second chapter.At present, additive perturbation and multiplicative perturbation are the two typesof the perturbation problems of the matrices. In this paper, we focus on the followingthree types of perturbation problems in addition sense:Firstly, we investigate the perturbation of the coefficient of matrix characteristicpolynomial considered from two perspectives, starting from the determinant expansion,using elementary symmetric function of the singular values and binomial coefficients todefine absolute perturbation bounds. Besides, we explore the normal matrix andHermite positive definite matrix situation. On the other hand，from the point of view ofthe eigenvalues, we study how to define the absolute and relative perturbation bounds ofthe coefficient of matrix characteristic polynomial by the elementary functions with res-pect to eigenvalues.Then we study the lower bound of minimum eigenvalue of the Hadamard productabout the M-matrix and nonnegative matrix, then further explore the lower bound ofminimum eigenvalue of the Hadamard product about the M-matrix and inverseM-matrix, to characterize the perturbation bounds by the matrix elements themselvesmainly, and give numerical examples to compared with previous conclusions, illustratethe conclusions are better than previously.Finally, we study the upper bound for the spectral radius of the Hadamard product of two nonnegative matrices, and obtain a new upper bound. It has been proved betterthan before result theoretically. Also give a numerical example to illustrate.