I am reading the following paper, which describes a technique based on image reconstruction with PCA:
Luis Malagón-Borja, Olac Fuentes, Object detection using image reconstruction with PCA, Image and Vision Computing, Volume 27, Issue 1, 2009, Pages 2-9, ISSN 0262-8856, http://dx.doi.org/10.1016/j.imavis.2007.03.004.
The relevant step is where the reconstruction error is described. Given an image $u$, its projection $p$ onto a set of principal components $P$ (computed from an image set) is given by $p = P(u - \mu)$, where $\mu$ is the mean of the image $u$ as a vector. Thus, its reconstruction $u'$ is given by: $$u' = P^{T}p + \mu = P^{T}P(u - \mu) + \mu.$$ What I am having trouble understanding is the following. If P is the set of principal components obtained via PCA, it is also an orthogonal matrix. If P is orthogonal, wouldn't it be the case that $P^{T}P=PP^{T}=I$, and consequently $u' = u$? I am sure that I am interpreting something wrong, since this would imply a perfect reconstruction no matter what dataset P was computed from.
$P$ is the matrix whose columns are the first $k$ eigenvectors of $C$.
Hence, $P \in \mathbb{R}^{ rc \times k}$
Hence $P^TP =I_k \in \mathbb{R}^{k \times k}$.
Also $PP^T \in \mathbb{R}^{rc \times rc}$, if $k < rc$, this is a singular matrix.
Remark: I do think the authors are confused about $PP^T$ and $P^TP$.
The matrix multiplication $p=P(u - \mu)$ does not makes sense as the size is not compatible.