Gram matrix

For a finite family of vectors $v_1,\dots ,v_n$ in an inner product space, its Gram matrix records all pairwise inner products:Its diagonal entries are the squared norms of the vectors, and its off-diagonal entries describe their pairwise alignment. For example, $v_1=(1,0)$ and $v_2=(1,1)$ in ${\mathbb{R}}^2$ giveThe positive definiteness of a Gram matrix relates the resulting quadratic form to the squared norm of a linear combination. In particular, $G$ is a positive semidefinite matrix, and it is invertible exactly when the original vectors are linearly independent. This permits geometric questions about a family to be answered through one matrix.

Two finite families with the same Gram matrix are related by a linear isometry between their linear spans: send ${\sum }_i{c}_i{v}_i$ to the same linear combination of the second family. Equality of the Gram matrices preserves squared norms, so every zero combination maps to zero and the rule is well-defined. If the two ambient inner product spaces are finite-dimensional and have the same dimension, completing orthonormal bases extends this map to an ambient unitary map. Equality of the ambient dimensions is needed for that last extension.