Gram matrix

[[concept]]

Topics

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Gram Matrix

Let $v_1,\dots ,v_n\in {\mathbb{F}}^k$ be a (finite) collection of vectors in ${\mathbb{F}}^k$, which has inner product $⟨\cdot ,{\cdot }^{\prime }⟩$. The Gram Matrix $G$ of the $v_i$ is given by $G_{ij}=⟨v_i,v_j⟩,G=V^{\ast }V$ Where $V$ is the matrix with $v_i$ as its $i$th column

Notes

• The lengths of the $v_i$ are the entries along the diagonal
• The angles between each pair are on the off-diagonal

This information describes the entire geometry of the set of vectors; If any other set of vectors $u_1,\dots ,u_n$ has the same Gram matrix, then there must exist some $Q\in \mathcal{O}(k)$ such that $Q{v}_i=u_i$ for all $i$.

In this case, it is possible to get from one point cloud to the other via some orthogonal transformation - this is an if and only if!

References

References

Wikipedia

See Also

Mentions

Mentions

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