One approach to compute the roots of a zero-dimensional polynomial system is based on resultant formulations and can be performed with floating point arithmetic. In this paper, reinvestigating the resultant approach from the linear algebra point of view the author handles the problem of genericity and presents a new algorithm for computing the isolated roots of an algebraic variety. The author analyses two types of resultant formulations, transforms them into eigenvector problems, and describes special linear algebra operations on the matrix pencils in order to reduce the root computation to a non-singular eigenvector problem. This new algorithm, based on pencil decompositions, has a good complexity even in the non-generic situations and can be executed with floating point arithmetic.