The spectrum of a modified linear pencil

Suppose the spectrum of a symmetric definite linear pencil is known. This paper addresses the question of what can be said about the spectrum when scalar multiples of a rank-one update are added to each matrix in the pencil. The secular equation for this problem is derived, and from it, a certain separation property is found which gives insight into the connection between the eigenvalues before and after modification. In the context of structural dynamics, the result characterises the behaviour of a finite-dimensional vibrating system undergoing mass and stiffness modifications. The result also leads to applications such as a divide and conquer algorithm for the eigenvalues of the modified system (so-called matrix tearing) and spectral shifting. An illustrative example is also given.