Suppose I have I have $m$ normalized examples $x$ in $d$ dimensions and $m\times m$ Gram matrix G where $$G_{ij}=\langle x_i, x_j\rangle$$

Is there a name or statistical/geometric interpretation of the following quantity?

$$\sum_{ij} G_{ij} G_{ji}$$

This seems to measure how far the examples are from being mutually orthogonal, ranging from $d$ for perfectly orthogonal set to infinity for a correlated, wondering if this comes up in literature.

Motivation: reciprocal of this quantity is proportional to drop in expected error norm squared after applying one step of gradient descent on least squares objective on dataset with examples $x$

Suppose I have I have vectors $x_1,\ldots,x_m$ in $\mathbb{R}^d$, with $\|x_i\|=1$ for all $i$

Is there a name or statistical/geometric interpretation of the following quantity?

$$\sum_{ij} \langle x_i, x_j \rangle ^2 $$

This seems to measure how far the examples are from being mutually orthogonal, evaluating to $m$ for perfectly orthogonal set of $x$'s. Wondering if this comes up in literature.

Motivation: reciprocal of this quantity is proportional to drop in expected error norm squared after applying one step of gradient descent on least squares objective on dataset with examples $x$