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A $G$-invariant inner product is constructed by averaging an arbitrary initial inner product $\langle \cdot, \cdot \rangle$ over the group orbit. The new inner product is defined as $(v, w) := \frac{1}{|G|} \sum_{g \in G} \langle \rho(g)v, \rho(g)w \rangle$.

Conditions: $G$ is a finite group.; $\rho: G \to GL(V)$ is a representation.; $\langle \cdot, \cdot \rangle$ is an arbitrary initial inner product on $V$.; $|G|$ is the order (cardinality) of the finite group $G$.

A $G$-invariant inner product is constructed by averaging an arbitrary initial inner product $\langle \cdot, \cdot \rangle$ over the group orbit. The new inner product is defined as $(v, w) := \frac{1}{|G|} \sum_{g \in G} \langle \rho(g)v, \rho(g)w \rangle$.

Conditions: $G$ is a finite group.; $\rho: G \to GL(V)$ is a representation.; $\langle \cdot, \cdot \rangle$ is an arbitrary initial inner product on $V$.; $|G|$ is the order (cardinality) of the finite group $G$.