= Characterization of linear surface-area isometries
{title2=$\operatorname{Area}(AS)=\operatorname{Area}(S)\ \forall S\ \Longleftrightarrow\ A^TA=I$}
For a real three-dimensional <linear map> $A$, preservation of every <surface area> is equivalent to being an <orthogonal matrix>. For necessity, diagonalize the <symmetric matrix> $A^TA$ in an <orthonormal basis>, with nonnegative <eigenvalues> $\lambda_i$. The three unit <coordinate> <parallelograms> have image <areas> $\sqrt{\lambda_1\lambda_2}$, $\sqrt{\lambda_1\lambda_3}$, and $\sqrt{\lambda_2\lambda_3}$. Setting all three equal to $1$ forces every $\lambda_i=1$. Sufficiency follows from preservation of <dot products> and $\|\mathbf u\times\mathbf v\|^2=\|\mathbf u\|^2\|\mathbf v\|^2-(\mathbf u\cdot\mathbf v)^2$. This specifically uses dimension three; in dimension two, preservation of planar <area> alone only requires $|\det A|=1$.
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