Exercise 1024Linear algebra › Inner product spacesL1Added 3 Nov 2026

Orthogonal matrices and isometries

#orthogonal matrix#isometry#inner product preservation

Statement

Let $Q$ be a real $n\times n$ matrix. Show that $Q$ is orthogonal ( $Q^TQ=I$ ) iff $\langle Qu,Qv\rangle=\langle u,v\rangle$ for all $u,v$ .

Hint

Compute $\langle Qu,Qv\rangle=u^T Q^T Qv$ .

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Solution

If $Q^TQ=I$ . Then $\langle Qu,Qv\rangle=(Qu)^T(Qv)=u^TQ^TQv=u^Tv=\langle u,v\rangle$ .

Conversely. If $\langle Qu,Qv\rangle=\langle u,v\rangle$ for all $u,v$ , then $u^T(Q^TQ-I)v=0$ for all $u,v$ . Taking $u=e_i$ , $v=e_j$ , we get $(Q^TQ)_{ij}=\delta_{ij}$ , so $Q^TQ=I$ .

Conclusion. Orthogonal matrices are exactly the linear isometries of $\mathbb{R}^n$ .

Consequences. $\det Q=\pm1$ , and the columns of $Q$ form an orthonormal basis.

Orthogonal matrices and isometries

Statement

Solution

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