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Equation 8 · Part 7 · How Llama's Architecture Actually Works, Generation by Generation

=

x2i−1′=x2i−1cos⁡(mθi)−x2isin⁡(mθi)x2i′=x2i−1sin⁡(mθi)+x2icos⁡(mθi)\begin{aligned} x'_{2i-1} &= x_{2i-1}\cos(m\theta_i) - x_{2i}\sin(m\theta_i) \\ x'_{2i} &= x_{2i-1}\sin(m\theta_i) + x_{2i}\cos(m\theta_i) \end{aligned}
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

and a vector at sequence position m has its i -th pair rotated by angle mθi\theta_i : x2i−1′=x2i−1cos⁡(mθi)−x2isin⁡(mθi)x2i′=x2i−1sin⁡(mθi)+x2icos⁡(mθi)\begin{aligned} x'_{2i-1} &= x_{2i-1}\cos(m\theta_i) - x_{2i}\sin(m\theta_i) \\ x'_{2i} &= x_{2i-1}\sin(m\theta_i) + x_{2i}\cos(m\theta_i) \end{aligned}. The property that makes this useful rather than merely elegant is that rotating a query at position m and a key at position n by these matrices before taking their dot product leaves a result that depends only on the distance between them:

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.