Is Principal Component Analysis a parametric method? Principal component analysis assumes that the features are distributed by a Gaussian distribution. Does this make Principal Component Analysis a parametric approach?
I can't seem to find a concrete answer saying that PCA is indeed a parametric approach. As we know, that parametric approaches assume some distribution of the input data.
 A: We do not have to make any distributional assumptions in order to extract the Principal Component directions from a covariance or correlation matrix. To see this, recall the problem PCA solves. For a $p\times p $ covariance matrix $\mathbf{\Sigma}$ and all vectors $\mathbf{x} \in \mathbb{R}^p$ we would like to maximize 
$$\mathbf{x}^{\prime} \mathbf{\Sigma} \mathbf{x} \tag{1}$$
subject to the condition that $\mathbf{x}$ is a unit vector, i.e. $\mathbf{x}^\prime \mathbf{x} = 1$. The solution is the eigenvector of the covariance (or correlation) matrix corresponding to the largest eigenvalue. The second principal component is the vector that maximizes $(1)$ subject to the additional condition that is perpendicular to the first direction and likewise for the third, forth and millionth principal component (always $\leq p$) . All we require is that the maximizers are unit vectors and perpendicular to the previous directions.
We do not need normality for the extraction but we definitely need the normality for hypothesis testing, e.g. to see how many directions are significant. It's worth noting, however, that with normality we have a pretty neat interpretation of PCA as the axes in ellipsoids of constant density (recall the exponent of a multivariate Normal distribution).
A: PCA has nothing to do with the normality of your data. The principle is that you have a bunch of data points in a (high-dimensional) space and you want to see which directions or principal vectors can describe your data in an optimal way. It is virtually the same as the singular value decomposition and can be applied to any matrix (here the matrix of your data points).
