How to reconstruct the design matrix using the principal score and loading in PCA

Question

I was reading the principal component analysis section of ISLR. The authors define the $m$th principal component score from the $m$th principal loading vector $\phi_1=(\phi_{1m},...,\phi_{pm})$ as $Z_m=\phi_{1m}X_1+...+\phi_{pm}X_m$. On page 387-388 of the pdf, the author states that for an $n$ by $p$ (observations by predictor) design matrix $X$ that is centered, if $M = min(n-1,p)$ (number of principal components where the score has non-zero variance), then one can reconstruct the original design matrix $X$ using $x_{ij}=\sum_{m=1}^M z_{im}\phi_{jm}$ (where $1 \leq i \leq n$ and $1 \leq j \leq p$) which is similar to this answer.

I'm trying to get an intuition as to why we only need the principal components where the score has non-zero variance to reconstruct the original design matrix.

Answer 1

The principal component scores along any principal component loading vector average to 0 (because of centering - see page 384 of the pdf). If the variance is also 0, then all the scores must also be 0. Hence we can take only the principal components with nonzero variance score vector.