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.
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 $\alpha_i\alpha_j^T = 0$.