Gram matrix
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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[1], p. 441[2]
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) \(\displaystyle \operatorname {rank} G=\dim \operatorname {span} \{v_{1},\ldots ,v_{m}\}\)
- ↑ Horn, Roger A.; Johnson, Charles R. (2013). "7.2 Characterizations and Properties". Matrix Analysis (Second Edition). Cambridge University Press. ISBN 978-0-521-83940-2.
- ↑ https://en.wikipedia.org/wiki/Gramian_matrix#cite_note-1