= Orthogonalization of a transplantation matrix
{title2=$O=C(C^TC)^{-1/2}$}
Suppose an invertible real matrix $C$ intertwines symmetric orthogonal tile-reflection matrices: $CM_s=N_sC$. Then $C^TC$ commutes with every $M_s$, so its positive inverse square root does too. The <polar decomposition of an invertible real matrix> therefore gives an orthogonal matrix $O=C(C^TC)^{-1/2}$ satisfying $OM_s=N_sO$. On a disjoint union of congruent tiles, $O$ preserves both the $L^2$ <inner product> and <Dirichlet energy>. The intertwining identities preserve the face trace conditions defining the assembled <Sobolev space>, so $O$ gives a <unitary equivalence> of the assembled <Laplacians>.
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