= Left stretch tensor
{title2=$V=(FF^T)^{1/2}$}
The <left stretch tensor> is $V=(FF^{\mathsf T})^{1/2}$, 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 $V=RUR^{\mathsf T}$. The two tensors therefore have the same <eigenvalues>, the <principal stretches>, but their <eigenvectors> belong to differently oriented configurations. For $F=R\operatorname{diag}(2,1,1/2)$, $V=R\operatorname{diag}(2,1,1/2)R^{\mathsf T}$. 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.
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