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Consider a set of vectors $S = \{u_1, \dots, u_k\}$ in $\mathbb{R}^n$. (I'm generally interested in scenarios when $k << n$, not sure how much this matters.)

Let $G_k$ denote the Gram matrix of the vectors in $S$, and let $G_{(k-1)}$ denote the Gram matrix of the vectors $S\setminus\{u_k\}$. Is there a way to relate $G_k^{-1}$ to $G_{(k-1)}^{-1}$ in any way? I notice that for the family of vectors I'm considering the topleft $(k-1) \times (k-1)$ block of $G_{k}^{-1}$ is numerically close to $G_{(k-1)}^{-1}$. Is there a way to quantify this closeness?

gen
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  • Technical term "augmented matrix". See here – Jean Marie Sep 09 '22 at 19:57
  • I doubt the numerical closedness. For example Let $0<a<1$ and $$G_2=\begin{pmatrix} 1 & a\ a&1\end{pmatrix}$$ Then $$G_2^{-1}= {1\over 1-a^2}\begin{pmatrix} 1 & -a\ -a&1\end{pmatrix}$$ The quantities $1$ and $(1-a^2)^{-1}$ are far apart if $ a\to 1^-.$ – Ryszard Szwarc Sep 11 '22 at 16:38

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