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 primarily used for dimensionality reduction while preserving as much variability as possible in the dataset. The key purpose of PCA is to identify patterns in high-dimensional data and express the data in such a way that it highlights their similarities and differences. It achieves this by transforming the original correlated features into a set of new uncorrelated features known as principal components. These new features are ordered by their variance, with the first principal component explaining the highest variance, the second explaining the next highest, and so on.

To understand how PCA works in terms of extracting principal components, consider a dataset with two variables (X1 and X2). The first step is to calculate the covariance matrix of the dataset. The covariance matrix is a square matrix that represents the covariance (i.e., ____ __________ _______ ____ _______ ______ _________ __________ __________.
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