A reduced form for perturbed matrix polynomials
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Computing eigenpairs of quadratic eigensystems
Mathematical and Computer Modelling: An International Journal
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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 $A(\lambda)=\lambda^2I+\lambda C+K$ 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 $t(\lambda,\epsilon)=det(A(\lambda)+ \lambda\epsilon C)$. (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 $\lambda_0$.