In very different contexts of mathematical physics (rigid body mechanics, fluidodynamics, general relativity, quantum field theory,...) I have come across the following expression: $$ A^tA-AA^t, $$
where $A$ is a square matrix and $A^t$ its transpose, or the vector analogue: $$ vv^t-\Bbb{1}v^tv, $$ where $\Bbb{1}$ is the identity and $v$ is a vector (or also the differential operator $\nabla$).
What do the quantities above really measure? Is there an intuitive, geometric or algebraic meaning? Any explanation which works only in one context (say, fluidodynamics) would be also welcome.
Thanks!
Addendum: Thanks to Autolatry's answer, here is an example of what I mean by "intuitive meaning": The first term alone, namely $A^tA$, has the significance of a magnitude (at least in the Euclidean case). It gives the metric tensor for embedded submanifolds, it gives the length of a vector, its determinant gives the volume form... I'm asking if the difference of the two terms above also has a geometrical meaning.
I can't think of a perfectly suitable description of what $AA^{T}$ and $A^{T}A$ can represent (certianly out of my comfort zone with regards fluid dynamics)other that to say that for some special types of groups the transpose matrix is actually the inverse, the rotation matrices have this property.So geometrically the rotation matrices preserve length.
A more rigorous geometric interpretation is to consider a parameterization of a $k$-manifold $M \in \mathbb{R}^{n}$ such that $\gamma: \mathbb{R}^{k} \rightarrow \mathbb{R}^{n}$.
The Jacobian $J=[D\gamma]$ is such that $J^{T}J$ is the metric induced on $M$ by the embedding in $\mathbb{R}^{n}$.