Low Rank Perturbation of Weierstrass Structure
Fernando De Terán, Froilán M. Dopico, Julio Moro · SIAM Journal on Matrix Analysis and Applications · 2008
Let $A_0 + \lambda A_1$ be a regular matrix pencil, and let $\lambda_0$ be one of its finite eigenvalues having g elementary Jordan blocks in the Weierstrass canonical form. We show that for most matrices $B_0$ and $B_1$ with ${\rm rank} (B_0 + \lambda_0 B_1)< g$ there are $g - {\rm rank} (B_0 + \lambda_0 B_1)$ Jordan blocks corresponding to the eigenvalue $\lambda_0$ in the Weierstrass form of the perturbed pencil $A_0+B_0 + \lambda (A_1+B_1)$. If ${\rm rank} (B_0 + \lambda_0 B_1)+ {\rm rank} (B_1)$ does not exceed the number of $\lambda_0$-Jordan blocks in $A_0 + \lambda A_1$ of dimension greater than one, then the $\lambda_0$-Jordan blocks of the perturbed pencil are the $g- {\rm rank} (B_0 + \lambda_0 B_1)-{\rm rank} (B_1)$ smallest $\lambda_0$-Jordan blocks of $A_0 + \lambda A_1$, together with ${\rm rank} (B_1)$ blocks of dimension one. Otherwise, all $g- {\rm rank} (B_0 + \lambda_0 B_1)$ $\lambda_0$-Jordan blocks of the perturbed pencil are of dimension one. This happens for any pair of matrices $B_0$ and $B_1$ except those in a proper algebraic submanifold in the set of matrix pairs. If $A_0 + \lambda A_1$ has an infinite eigenvalue, then the corresponding result follows from considering the zero eigenvalue of the dual pencils $A_1 + \lambda A_0$ and $A_1 + B_1 + \lambda (A_0 + B_0)$.