= Solution
A <Lorentz transformation> preserves the quadratic form of the <Minkowski metric> for every four-vector. Substituting $x'^\sigma=\Lambda^\sigma{}_{\mu}x^\mu$ gives
$$
\eta_{\sigma\tau}\Lambda^\sigma{}_{\mu}\Lambda^\tau{}_{\nu}x^\mu x^\nu
=\eta_{\mu\nu}x^\mu x^\nu.
$$
The coefficient difference is a symmetric matrix, and a symmetric quadratic form that vanishes on every vector is zero: evaluating it on basis vectors and their sums recovers each coefficient. Consequently
$$
\boxed{\Lambda^T\eta\Lambda=\eta,\qquad
\eta_{\mu\nu}=\eta_{\sigma\tau}\Lambda^\sigma{}_{\mu}\Lambda^\tau{}_{\nu}.}
$$
This is the defining metric-preservation relation of the <Lorentz group>.
Back to article page
