Gram matrix

Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $v_1, \dots, v_k \in V$. The Gram matrix of $v_1, \dots, v_k$ is the $k \times k$ matrix $G$ with entries $$G_{ij} = \langle v_i, v_j \rangle \qquad (1 \le i, j \le k),$$ where $\langle \cdot, \cdot \rangle$ is the inner product.

Remarks

The Gram matrix is Hermitian and positive semidefinite. It is invertible if and only if $v_1, \dots, v_k$ are linearly independent. For real column vectors $v_i \in \R^n$ with the dot product, $G = A^T A$, where $A$ has the columns $v_1, \dots, v_k$. Its determinant is the square of the $k$-dimensional volume of the parallelepiped spanned by the $v_i$.

Explanations

L1 human

The Gram matrix records all inner products of a list of vectors. It contains all lengths and all angles between them. Two lists with the same Gram matrix have the same shape, up to an isometry.