Applications of simultaneous diagonalization of quadratic forms

Question

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.

What else do you know about S? Is it orthogonal by chance? – N8tron May 05 '14 at 17:35 — N8tron, May 05 '14 at 17:35
@NateIverson: Not necessarily. – user35603 May 05 '14 at 18:35 — user35603, May 05 '14 at 18:35
I don't think it's diagonalization unless $S^T=S^{-1}$ – N8tron May 06 '14 at 00:22 — N8tron, May 06 '14 at 00:22

score 2 · Accepted Answer · answered May 06 '14 at 17:03

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.

Applications of simultaneous diagonalization of quadratic forms

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