Questions on PCA: when are PCs independent? why is PCA sensitive to scaling? why are PCs constrained to be orthogonal?

I am trying to understand some descriptions of PCA (the first two are from Wikipedia), emphasis added:

Principal components are guaranteed to be independent only if the data set is jointly normally distributed.

Is the independence of principal components very important? How can I understand this description?

PCA is sensitive to the relative scaling of the original variables.

What does 'scaling' mean there? Normalization of different dimensions?

The transformation is defined in such a way that the first principal component has the largest possible variance and each succeeding component in turn has the highest variance under the constraint that it be orthogonal to the preceding components.

Can you explain this constraint?

• #2 only applies if PCA is performed by eigendecomosition of the covariance matrix. If it is performed by eigendecomposition of the correlation matrix, then PCA is insensitive to scaling. – Alexis Aug 3 '14 at 22:42
• @Alexis Thank you for your post. For #2, would you mind to explain what does the'scaling' mean? the dynamic change of the corresponding dimension of data? – kakanana Aug 3 '14 at 22:46
• "Scaling" can mean a few things. (1) It can mean linear transformations of data $\mathbf{X}$, such as $\mathbf{X^{*}} = a + b\mathbf{X}$, where $-\infty < a < \infty$ and $0 < b < \infty$; or (2) that the individual variables in $\mathbf{X}$ are all measured on the same scale and have closely sized variances. My comment applies to both of these meanings. – Alexis Aug 4 '14 at 1:33

Q3. The "constraint" is how PCA works (see a huge thread). Imagine your data is 3-dimensional cloud (3 variables, $n$ points); the origin is set at the centroid (the mean) of it. PCA draws component1 as such an axis through the origin, the sum of the squared projections (coordinates) on which is maximized; that is, the variance along component1 is maximized. After component1 is defined, it can be removed as a dimension, which means that the data points are projected onto the plane orthogonal to that component. You are left with a 2-dimensional cloud. Then again, you apply the above procedure of finding the axis of maximal variance - now in this remnant, 2D cloud. And that will be component2. You remove the drawn component2 from the plane by projecting data points onto the line orthogonal to it. That line, representing the remnant 1D cloud, is defined as the last component, component 3. You can see that on each of these 3 "steps", the analysis a) found the dimension of the greatest variance in the current $p$-dimensional space, b) reduced the data to the dimensions without that dimension, that is, to the $p-1$-dimensional space orthogonal to the mentioned dimension. That is how it turns out that each principal component is a "maximal variance" and all the components are mutually orthogonal (see also).