Gram matrix

Gram matrix \( G = V^T V\) where \(V\) is a matrix.(for finite-dimensional vectors with the usual Euclidean dot product) [1]

Horn & Johnson 2013, p. 441
Theorem 7.2.10 Let \(\displaystyle v_{1},\ldots ,v_{m}\) be vectors in an inner product space V with inner product \(\displaystyle \langle {\cdot ,\cdot }\rangle \) and let \(\displaystyle G=[\langle {v_{j},v_{i}}\rangle ]_{i,j=1}^{m}\in M_{m}\). Then
(a) G is Hermitian and positive-semidefinite
(b) G is positive-definite if and only if the vectors \(\displaystyle v_{1},\ldots ,v_{m}\) are linearly-independent.
(c) \(\displaystyle \operatorname {rank} G=\dim \operatorname {span} \{v_{1},\ldots ,v_{m}\}\)

(b) 는 \( x^TGx = x^TV^TVx = (Vx)^T(Vx) \ge 0 \) 로 하면 될것 같고, (c)는 \(x^TGx =0\)인 상황에서 \(V\)가 nonsingular하면 모순임을 이용하면 될 것 같다.

Gram determinant는 위키 참조.
(주요 내용은 1) volume of perelloptope 2) exterior product : \(G(x_{1},\dots ,x_{n})=\|x_{1}\wedge \cdots \wedge x_{n}\|^{2}\). )

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