Is MCA equivalent to PCA when all variables are binary?

I am looking to apply principal component analysis on binary (true/false) data, and I have come across the "equivalence between PCA and MCA" (Multiple Correspondense Analysis) for binary data, but haven't been able to find a reference to cite or check the proof.

For example the comment by IHateDerekBeaton in the thread below suggests they are the same for binary data: https://www.reddit.com/r/statistics/comments/3hq2oq/pca_or_equaivalent_on_sparse_binary_matrix/

Similarly, the comment by ttnphns in the question below suggest they are equivalent Would PCA work for boolean (binary) data types?.

So is there any proof for this or any reference that can be cited?

• You may find it out that MCA is a particular case of CatPCA (categorical PCA): if all variables are requested the "multiple nominal" (i.e. dummy variable) quantification then CatPCA becomes MCA. Further thing, if all variables are dichotomous then it doesn't matter which quantification type you select in CatPCA: it "degenerates" right into MCA. Now, CatPCA with such (binary) variables is virtually equal to usual PCA (slight numerical differences might be due to different algorithms, and CatPCA is iterative). – ttnphns Aug 30 '18 at 13:26
• See answers stats.stackexchange.com/a/34878/3277; stats.stackexchange.com/a/200917/3277. Read about variants of algorithms of MCA (giving the same result) in Wikipedia and also in documentation.statsoft.com/…, documentation.statsoft.com/… – ttnphns Aug 30 '18 at 13:29
• Two last links seem broken, alas. – ttnphns Aug 30 '18 at 13:39
• Thanks @ttnphns for the info, so from what i understand , if I apply PCA on a matrix where true is represented by 1 and false by 0, then I am essentially doing MCA (ignoring numerical difference due to the nature of the algorithm)? – reverb Aug 30 '18 at 15:02
• @ttnphns Also, if the above is true, does standardizing the binary data (e.g. making each column zero-mean) have any effect on this equivalence? – reverb Aug 30 '18 at 15:07