= Right stretch tensor
{title2=$U=(F^TF)^{1/2}$}
The <right stretch tensor> is $U=(F^{\mathsf T}F)^{1/2}$, the positive <matrix square root> of the <right Cauchy-Green deformation tensor>. In the <polar decomposition in continuum mechanics> $F=RU$, it stretches material directions in the <reference configuration> before the <rotation> $R$ carries them to the final orientation.
For a unit reference direction $\mathbf N$, the squared length ratio is $|F\mathbf N|^2=\mathbf N\cdot U^2\mathbf N$. Along an <eigenvector> of $U$, that ratio is the square of its <principal stretch>. For $F=R\operatorname{diag}(2,1,1/2)$, one has $U=\operatorname{diag}(2,1,1/2)$ regardless of $R$: the local lengths double, stay fixed and halve along the three reference axes. Since the product is one, this example preserves <volume>. The <Biot strain tensor> $U-I$ expresses these extension ratios relative to unity.
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