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If so, why would you apply another rotation after you already found the "variance-maximizing" - and in that sense optimal - rotation. Wouldn't the second rotation lead to a - again - non-variane-maximizing situation?

Please enlight me.

PeterPancake
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    Do you know practical cases in which all the components are kept? I believe that the second rotation is most often done after eliminating (filtering) the smaller components (slash selecting the larger components) and is most often done for the purpose of easier representation (aligning with more meaningful conceptual axes). – Sextus Empiricus Jan 19 '18 at 13:56
  • What you mean by 'second rotation' and 'optimal rotation' is unclear. – Sextus Empiricus Jan 19 '18 at 13:57
  • PCA is just the singular value decomposition for a data matrix. The gyst of SVD is that any matrix (using your language) can be factored as a 'first rotation', followed by a change of scale followed by a second rotation. So any linear change of variables is just a rotation followed by a scale change. – meh Jan 19 '18 at 14:12
  • So far, I understand PCA with "all components" as a rotation, as my question suggests (= first & optimal rotation) – PeterPancake Jan 19 '18 at 14:14
  • There's an internal contradiction in the question. Keeping all $n$ PCs indeed corresponds to a rotation which is (almost) unique. But the "variance maximizing rotation" is far from unique: it merely identifies one direction out of an $n-1$ dimensional manifold of possible directions, thereby determining an $(n-1)(n-2)/2$ dimensional manifold of possible rotations. When $n=2$ the first PC determines both PCs, but for $n\gt 2$ the first PC does not fully determine a rotation. – whuber Jan 19 '18 at 15:57

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