Wavelets and Linear Algebra
E. Kokabifar, Ghasem Barid Loghmani, Ali Mohammad Nazari, Seyed Mahdi Karbassi · 2015
Consider an nn matrix polynomial P( ). A spectral norm dis- tance from P( ) to the set of nn matrix polynomials that have a given scalar 2 C as a multiple eigenvalue was introduced and obtained by Papathanasiou and Psarrakos. They computed lower and upper bounds for this distance, constructing an asso- ciated perturbation of P( ). In this paper, we extend this result to the case of two given distinct complex numbers 1 and 2. First, we compute a lower bound for the spectral norm distance from P( ) to the set of matrix polynomials that have 1; 2 as two eigenvalues. Then we construct an associated perturbation of P( ) such that the perturbed matrix polynomial has two given scalars 1 and 2 in its spectrum. Finally, we derive an upper bound for the distance by the constructed perturbation of P( ). Numerical examples are provided to illustrate the validity of the method.