# Physical applications of matrix congruence?

## Main Question or Discussion Point

Just curious, are there any?

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HallsofIvy
Homework Helper
What do you mean by "matrix congruence"? Congruent in what sense?

Consider, for example, the diagonalization of the Moment of Inertia tensor, to find the principal moments of inertia and the respective principal axis of rotation.

Consider, for example, the diagonalization of the Moment of Inertia tensor, to find the principal moments of inertia and the respective principal axis of rotation.
But in that case similarity transformation would be enough, it happens to be a congruence transformation just because inertia tensor is real symmetric. So nothing nontrivial about congruence relation is actually invoked.

Yes, but the fact that this particular tensor (and many other 2nd order ones) is symmetric is a nontrivial fact, a consequence of deeper physical laws. What kind of application would you consider nontrivial?

In the inertia tensor case we only need to treat it as similarity transformation, it'll work just fine. No special properties of a congruence transformation are used, so I don't think it's a application of congruence transformation.

I disagree: a similarity transformation does not have to preserve the orthogonality of the reference frame axis', while a congruence (which is a particular case) does. The existence of the principal axis of rotation and moments of inertia depends on that preservation property.

As you said, only in a particular case congruence preserve the orthogonality, then why not just view it as a particular case of similarity transformation? It's not fair you compare the special case of congruence with the general case of similarity.