A COMPLETE PAIR OF SOLVENTS OF A QUADRATIC MATRIX PENCIL

The paper proves two key equivalences: (i) X is a right solvent of λ^2I+λB+C iff the n-dimensional graph subspace span([I;X]) is invariant under the companion matrix C1 (Theorem 16), and (ii) X,Z form a complete pair (X−Z invertible) iff the two graph subspaces span([I;X]) and span([I;Z]) are C1-invariant and give a direct-sum decomposition C^{2n}=M1⊕M2 (Theorem 18). The candidate solution reproduces these statements: Lemma 1 matches Theorem 16 exactly, and Lemma 2 recasts the paper’s rank argument for Theorem 18 via an explicit block-column operation on W=[I I; X Z], yielding WQ=[I 0; X Z−X], hence W invertible ⇔ X−Z invertible. This is the same linear-algebraic core of the paper’s proof, just streamlined. Thus both are correct and essentially the same proof path ; the definition of “complete pair” (X−Z invertible) is the same in both .