detIn

DetIn is a term used in linear algebra to denote the determinant of the Gram matrix associated with a set of vectors in an inner product space. If v1, v2, ..., vn are vectors in a real or complex inner product space V with inner product ⟨·,⟩, the Gram matrix G is defined by Gij = ⟨Vi, Vj⟩. The quantity detIn(v1, ..., vn) is then defined as det(G). This value equals the squared n-dimensional volume of the parallelepiped spanned by the vectors.

Formally, detIn is the determinant of the matrix of inner products among the chosen vectors, sometimes also described as the Gram determinant. It is a nonnegative real number in real spaces and vanishes precisely when the vectors are linearly dependent. The quantity is invariant under orthogonal (or unitary) transformations of the vector set: applying an orthogonal change of basis leaves detIn unchanged. If all vectors are scaled by a common factor c, detIn scales by c^{2n}. When n exceeds the dimension of the space, the Gram matrix is singular and detIn = 0.

Example: In R^2, take v1 = (1, 0) and v2 = (1, 1). The Gram matrix is G = [[1, 1], [1, 2]], so detIn(v1, v2) = det(G) = 1. This matches the squared area of the parallelogram spanned by the two vectors.

DetIn is not universally standardized; in many discussions this quantity is referred to as the Gram determinant. It relates closely to the wedge product in exterior algebra and to various measures of linear independence and geometric volume in Euclidean space.