  Principal Component Analysis (PCA) is a transform for finding the principal
components (PCs) that represent features of random data. PCA also provides a
reconstruction of the PCs to the original data. We consider an extension of PCA
which allows us to improve the associated accuracy and diminish the numerical
load, in comparison with known techniques. This is achieved due to the special
structure of the proposed transform which contains two matrices $T_0$ and
$T_1$, and a special transformation $\mathcal{f}$ of the so called auxiliary
random vector $\mathbf w$. For this reason, we call it the three-term PCA. In
particular, we show that the three-term PCA always exists, i.e. is applicable
to the case of singular data. Both rigorous theoretical justification of the
three-term PCA and simulations with real-world data are provided.
