Some Properties Of Numerical Range, Quadratic Numerical Range And Spectrum Of Matrices

Relations between the quadratic numerical range and the spectrum of normal matrix are investigated, by using the unitary transformation and techniques of block matrix. Based on the special properties of normal matrix, an equality between the spectrum and the quadratic numerical range of normal matrix is established, which implies a new description of the spectrum of normal matrix. Besides, by means of the inclusion property of spectrum and the numerical range of product of matrices, a more weaker condition is given for judging a matrix to be a scalar multiple of a positive semidefinite matrix, and some existing results are improved.