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I am dealing with orthogonal matrices in regressions, so every regressor has

$x_j'x_k=0, if \ j \neq k \\ x_j'x_k=1, if \ \ j=k $

The first one told us that the correlation between two predictors is zero, but what told us the 2nd requirement?

It should something similiar as an standardization where all predictors ar on the same scale. But the latter requirement should differ a bit from standardization (in case of Interpretation).

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The second equation tells you that the feature (or covariate) has variance (or standard deviation) 1.

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