This tutorial on Principal Component Analysis (PCA) aims to provide a comprehensive understanding of the technique, focusing on both intuition and mathematical rigor. PCA is a widely used method in modern data analysis, particularly in fields like neuroscience and computer graphics, for reducing complex datasets to simpler, lower-dimensional structures. The tutorial begins with a motivating example of tracking a ball's motion using multiple cameras, illustrating the need for a more efficient representation of the data. It then introduces the concept of PCA as a change of basis, emphasizing the importance of linearity and the goal of maximizing variance while minimizing redundancy. The tutorial explains that PCA can be understood through the lens of linear algebra, specifically through eigenvector decomposition and singular value decomposition (SVD). It derives the principal components as the eigenvectors of the covariance matrix, which are found by solving a system of linear equations. The tutorial also discusses the limitations of PCA, such as its inability to handle non-linear relationships and non-Gaussian data, and introduces alternative methods like kernel PCA and independent component analysis (ICA) for more complex scenarios. Finally, the tutorial provides a step-by-step guide on how to implement PCA, including subtracting the mean from the data and calculating the eigenvectors or singular values. It concludes with a discussion on the strengths and weaknesses of PCA, emphasizing its non-parametric nature and the importance of understanding its underlying assumptions.This tutorial on Principal Component Analysis (PCA) aims to provide a comprehensive understanding of the technique, focusing on both intuition and mathematical rigor. PCA is a widely used method in modern data analysis, particularly in fields like neuroscience and computer graphics, for reducing complex datasets to simpler, lower-dimensional structures. The tutorial begins with a motivating example of tracking a ball's motion using multiple cameras, illustrating the need for a more efficient representation of the data. It then introduces the concept of PCA as a change of basis, emphasizing the importance of linearity and the goal of maximizing variance while minimizing redundancy. The tutorial explains that PCA can be understood through the lens of linear algebra, specifically through eigenvector decomposition and singular value decomposition (SVD). It derives the principal components as the eigenvectors of the covariance matrix, which are found by solving a system of linear equations. The tutorial also discusses the limitations of PCA, such as its inability to handle non-linear relationships and non-Gaussian data, and introduces alternative methods like kernel PCA and independent component analysis (ICA) for more complex scenarios. Finally, the tutorial provides a step-by-step guide on how to implement PCA, including subtracting the mean from the data and calculating the eigenvectors or singular values. It concludes with a discussion on the strengths and weaknesses of PCA, emphasizing its non-parametric nature and the importance of understanding its underlying assumptions.