Suppose I want to do a different version of PCA, in which I want to minimize the variances explained.
So lets say I find the first such variance minimizing axis $PC^{'}_1$. Then I impose the orthogonality constraint and find the second variance minimizing axis $PC^{'}_2$ and so on, to get $PC^{'}_1, PC^{'}_2,...,PC^{'}_n$.
And suppose the standard principal components in the descending order of variances explained are given by $PC_1, PC_2, ..., PC_n $.
So is it always true that $PC_i = PC^{'}_{n+1-i}$ for all $1 \leq i \leq n$?
If the answer is yes, then how can we prove it?
And if the answer is no, how can we compute the aforementioned decomposition?
I myself am not sure on how to go about proving/disproving this.