Euclidean vector space/Isometry/Orthogonal/Fact

Matrix characterization of an isometry

Let ${}V$ be a Euclidean vector space, and let ${}u_{1},\ldots ,u_{n}$ denote an orthonormal basis of ${}V$. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping, and let ${}M$ be the describing matrix of ${}\varphi$ with respect to the given basis.

Then ${}\varphi$ is an

isometry if and only if

${}{M^{\text{tr}}}M=E_{n}\,$

holds.

Proof, Write another proof