Linear and multilinear algebra; matrix theory Definition

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 $v_i$ .

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Vector space definition
Inner product definition
Gram matrix definition

Tag	`0HX8HZ1WTQ`
URI	`m2://0HX8HZ1WTQ`
Kind	definition
Writing	W1 stated
MSC	15A63, 15B57
Defines	Gram matrix
Uses	Inner product

Gram matrix

Definition

Remarks

Reading path

Further reading

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