If $A$ and $B$ are square symmetric matrices and, additionally, one of them, say $B$, is positively defined, then there exists an invertible matrix $S$ such that
$$S^{\top}\!AS=D \quad\text{and}\quad S^{\top}\!BS=I,$$
where $D$ is diagonal and $I$ is the identity matrix.
Question: Why it is important to be able to do such reduction simultaneously (by a single matrix $S$)? Where this can be applied?
P.S.: I heard about some applications for differential equations, but only in general phrases.
This is also known as the modal decomposition in engineering applications, specially in vibrations. It is very useful for the decoupling of differential equations.
A good example of its use in this context is given in pp.160-173 of Theoretical Mechanics of Particles and Continua, which preview is fortunately available in google books. In special, pp.163-173 illustrates the use of this technique for a scenario with two coupled pendulums.