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When we carry out PCA, is my understanding correct that the new principal components will need to agree on the same origin as the "original" axes of the data?

I have read through this great thread on PCA here: what confused me slightly is that the animations done by user Amoeba suggest the first PC does not need to go through the origin...

I always assumed that in linear algebra, whatever transformation we do, the origin has to remain the same (otherwise it's no longer linear...?)

Jan Stuller
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  • In that animation, all the PCs share a common origin. – whuber Jul 11 '21 at 15:54
  • @whuber: thank you. But do all the PCs have to share the same origin as the **original data** ? – Jan Stuller Jul 11 '21 at 15:55
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    The data are usually centered before calculating the PCA. In that case, the PCs always have the same origin, namely $(0,0)$. Maybe the pictures in [this post](https://stats.stackexchange.com/a/22331/21054) help in understanding. – COOLSerdash Jul 11 '21 at 16:20
  • See https://stats.stackexchange.com/questions/22329. https://stats.stackexchange.com/questions/485405 might also be helpful. The fact is that you're free to recenter any of the variables before performing PCA. – whuber Jul 11 '21 at 16:29
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    @whuber: thank you! That is very helpful. – Jan Stuller Jul 11 '21 at 16:54

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