  We consider a family of vector dot products that can be implemented using
sign changes and addition operations only. The dot products are
energy-efficient as they avoid the multiplication operation entirely. Moreover,
the dot products induce the $\ell_1$-norm, thus providing robustness to
impulsive noise. First, we analytically prove that the dot products yield
symmetric, positive semi-definite generalized covariance matrices, thus
enabling principal component analysis (PCA). Moreover, the generalized
covariance matrices can be constructed in an Energy Efficient (EEF) manner due
to the multiplication-free property of the underlying vector products. We
present image reconstruction examples in which our EEF PCA method result in the
highest peak signal-to-noise ratios compared to the ordinary $\ell_2$-PCA and
the recursive $\ell_1$-PCA.
