On Eigenvalues of Quadratic Matrix Polynomials and Their Perturbations
Mehdi Radjabalipour, Abbas Salemi · SIAM Journal on Matrix Analysis and Applications · 1996
Following the terminology used by Gohberg, Lancaster, and Rodman, the main results of the paper are as follows. (i) Studying the values of the partial multiplicities of a matrix polynomial $A( \lambda ) = \lambda ^2 I + \lambda C + K$ with hermitian coefficients at real eigenvalues $\lambda _0 $ and determining sharp bounds for the highest degree d of the factor $( \lambda - \lambda _0 )^d $ in the bivariate polynomial $t ( \lambda ,\epsilon ) = \det(A ( \lambda ) + \lambda \in C)$. (ii) Finding conditions on general matrices C and K implying that the leading exponent in the Puiseux expansion of the zero $\lambda ( \epsilon )$ of $t( \lambda ,\epsilon ) = 0$ near $\lambda _0 $ is $1 / a$, where a is the algebraic multiplicity of $\lambda _0 $.