I was reading some notes and it says that PCA can "sphere the data". What they define to me as "sphering the data" is dividing each dimension by the square root of the corresponding eigenvalue.
I am assuming that by "dimension" they mean each basis vector into which we are projecting (i.e. the eigenvectors we are projecting to). Thus I guess they are doing:
$$ u^{'}_i= \frac{u_i}{\sqrt{eigenValue(u_i)}}$$
where $u_i$ is one of the eigenvectors (i.e. one of the principal components). Then with that new vector, I am assuming they are projecting the raw data we have, say $x^{(i)}$ to $z^{(i)}$. So the projected points would now be:
$$ z'^{(i)} = u^{'}_i \cdot x^{(i)}$$
They claim that doing this ensures that all the features have the same variance.
However, I am not even sure if my interpretation of what they mean by sphering is correct and wanted to check if it was. Also, even if it was correct, what is the point of doing something like this? I know they claim it makes sure everyone has the same variance but, why would we want to do this and how does it achieve this?
u
is eigenvectors's value and is related to raw PC values.u'
is called a loading and is related to the normalized (equal variances) PC values. You may want read my answer about it: stats.stackexchange.com/a/35653/3277. $\endgroup$