# Inverse of a Gram matrix

If $$G$$ is the Gram matrix of $$n$$ vectors of size $$d$$ that are linearly independent ($$n\leq d$$), then $$G$$ is invertible and its inverse is a Gram matrix. Of what?

To be more precise, if $$G$$ is the Gram matrix of $$u_1,\ldots,u_n$$, where each $$u_i\in\mathbb R^d$$ and $$u_1,\ldots,u_n$$ are linearly independent, then there exists a collection of $$n$$ linearly independent vectors $$v_1,\ldots,v_n$$ whose $$G^{-1}$$ is the Gram matrix. Is there some relationship between the $$u$$'s and the $$v$$'s?

yes, $$\langle u_i, v_j \rangle = \delta_{ij}$$ meaning $$1$$ if $$i=j$$ but $$0$$ otherwise
The convention in Riemannian geometry is to write the elements of the original $$G$$ as $$g_{ij},$$ and the elements of the inverse matrix as $$g^{ij}.$$ The original basis is often $$e_i,$$ while the new basis is $$f^j.$$ The construction is $$f^i = \sum_j g^{ij} e_j$$