# Why "Right" and "Left" Cauchy-Green tensor?

$$C=F^TF$$ is called the "Right" Cauchy-Green tensor, and $$b=FF^T$$ is called the "Left" Cauchy-Green tensor.

I suppose in $$C=F^TF$$ the non-transposed $$F$$ stands on the right, and in $$b=FF^T$$ it stands on the left, but I guess there's gotta be more to it? Is there any reason for why they are called "right" and "left" specifically?

I would guess that it has to do with the polar decomposition of the deformation gradient ($$\mathbf{F}$$)
$$\mathbf{F} = \mathbf{R} \mathbf{U} = \mathbf{V}\mathbf{R}\, ,$$
with $$\mathbf{R}$$ an orthogonal tensor, $$\mathbf{U} = \mathbf{C}^{1/2}$$ the right stretch tensor, and $$\mathbf{V} = \mathbf{b}^{1/2}$$ the left stretch tensor.
Both, $$\mathbf{C}$$ and $$\mathbf{b}$$, should be positive definite tensors for the square root to be well defined.