Why do we maximize variance during Principal Component Analysis?

Question

I'm trying to read through PCA and saw that the objective was to maximize the variance. I don't quite understand why. Any explanation of other related topics would be helpful

Answer 1

Variance is a measure of the "variability" of the data you have. Potentially the number of components is infinite (actually, after numerization it is at most equal to the rank of the matrix, as @jazibjamil pointed out), so you want to "squeeze" the most information in each component of the finite set you build.

If, to exaggerate, you were to select a single principal component, you would want it to account for the most variability possible: hence the search for maximum variance, so that the one component collects the most "uniqueness" from the data set.

Answer 2

Note that PCA does not actually increase the variance of your data. Rather, it rotates the data set in such a way as to align the directions in which it is spread out the most with the principal axes. This enables you to remove those dimensions along which the data is almost flat. This decreases the dimensionality of the data while keeping the variance (or spread) among the points as close to the original as possible.