TITLE: Applications of simultaneous diagonalization of quadratic forms
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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.