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HandWiki. Gramian Matrix. Encyclopedia. Available online: https://encyclopedia.pub/entry/31485 (accessed on 22 September 2026).
HandWiki. Gramian Matrix. Encyclopedia. Available at: https://encyclopedia.pub/entry/31485. Accessed September 22, 2026.
HandWiki. "Gramian Matrix" Encyclopedia, https://encyclopedia.pub/entry/31485 (accessed September 22, 2026).
HandWiki. (2022, October 27). Gramian Matrix. In Encyclopedia. https://encyclopedia.pub/entry/31485
HandWiki. "Gramian Matrix." Encyclopedia. Web. 27 October, 2022.
Gramian Matrix
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The content is sourced from: https://handwiki.org/wiki/Gramian_matrix

In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors [math]\displaystyle{ v_1,\dots, v_n }[/math] in an inner product space is the Hermitian matrix of inner products, whose entries are given by [math]\displaystyle{ G_{ij}=\langle v_i, v_j \rangle }[/math]. An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram.

inner product space g_{ij}\langle linearly

References

  1. Lanckriet, G. R. G.; Cristianini, N.; Bartlett, P.; Ghaoui, L. E.; Jordan, M. I. (2004). "Learning the kernel matrix with semidefinite programming". Journal of Machine Learning Research 5: 27–72 [p. 29]. https://dl.acm.org/citation.cfm?id=894170. 
  2. Horn & Johnson 2013, p. 441, p.441, Theorem 7.2.10
  3. (Horn Johnson), p. 452, Theorem 7.3.11
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