What is the purpose of principal component analysis? Given the covariance matrix of order 2× ,2 explain how would you extract the principal components. Also, explain how would you find the proportion of total population variance for all the principal components.

Principal Component Analysis (PCA) is a statistical technique used in the field of probability and statistics for dimensionality reduction while preserving as much variability in the data as possible. The main objective of PCA is to transform a set of possibly correlated variables into a smaller number of uncorrelated variables called principal components. These components are ordered in such a way that the first few retain most of the variation present in all of the original variables. This is particularly useful when dealing with high-dimensional data, as it simplifies analysis and visualization, reduces noise, and helps in identifying patterns.

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