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Dont stop at EigenDecomp; re-use it or keep that pre-eigvenvector variable.

\[\lambda_{1, t} = \gamma_0 + \gamma_1 \beta_{\text{vol}, t} + \epsilon_t\]

Lambda being the eigenvalue of chosen from a data set with volatility.
two gammas being the baseline [gamma 0] and the other being sensitivity to that eigenvalue/voaltility [gamma 1].
epsilon being the error term.
*beta vol explained below*

The reason to not stop at Eigenvalue decomposition and re-use that eigenvalue is because in the process shown above in Latex, you can use the eigenvalue to fit into a linear regression of volatility -- which is what BetaVol is.

Try it out, its good to map momentum over a rolling window *given if your eigenvalue coems from volatility of course; if not the scalar wouldnt work*