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I've a question regarding pca variant.

Let $X ∈ \Bbb R^{D×n}$ be a data matrix, $\{u_i\}^{d}i=1$ be the $d$ principal components of $X$, and where $μ ∈ \Bbb R^d$ is the sample mean vector and $1_n ∈ \Bbb R^n$ is the $n$-dimensional ones vector.

We can define the new PCA based coordinates as $α_i = u^T_i(X − μ1^T_n ), i = 1, ..., d$.

can u explain why the new PCA features $α_i, α_j$ have zero mean and are uncorrelated.

joen joe
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  • I did my best to reformat your post, but I don't understand what {ui}di is supposed to mean. For future reference, see this page regarding how mathematical expressions are supposed to be formatted. – Ben Grossmann Mar 31 '22 at 15:08
  • i'vei change it. thanks for your response but i not understand yet – joen joe Mar 31 '22 at 15:45
  • Please explain what you don't understand. While I'm at it, note that askers are generally expected to provide context for their answers, as is explained here. So, please [edit] your question to say a bit about where you encountered this problem, what your thoughts are on this problem, and what you have tried so far. – Ben Grossmann Mar 31 '22 at 15:46

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Hint: A row-vector $v$ has mean zero iff it satisfies $v1_n^T = 0$. The lack of correlation between $\alpha_i,\alpha_j$ amounts to the observation that $\alpha_i\alpha_j^T = 0$.

Ben Grossmann
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