Left stretch tensor ($V=(F{F}^T{)}^{1/2}$)

The left stretch tensor is $V=(F{F}^{\mathsf{T}}{)}^{1/2}$, the positive matrix square root of the left Cauchy-Green deformation tensor. In the polar decomposition in continuum mechanics $F=VR$, it applies stretches along directions in the current configuration after the rotation $R$.

It is related to the right stretch tensor by $V=RU{R}^{\mathsf{T}}$. The two tensors therefore have the same eigenvalues, the principal stretches, but their eigenvectors belong to differently oriented configurations. For $F=R\mathrm{diag}(2,1,1/2)$, $V=R\mathrm{diag}(2,1,1/2)R^{\mathsf{T}}$. This distinction is useful when comparing a material-direction measurement before loading with an observed direction after the specimen has both stretched and turned. For a pure rotation, $V=I$, correctly reporting no local change in length.