This tutorial provides an intuitive and mathematical explanation of Principal Component Analysis (PCA), a widely used technique in data analysis. PCA is a method for reducing the dimensionality of data while preserving as much of the original information as possible. The goal of PCA is to identify the most meaningful basis for re-expressing a dataset, which can help reveal hidden structures and filter out noise. The tutorial begins with a simple example involving the motion of a spring, illustrating how PCA can be used to extract the underlying dynamics from noisy measurements. It then introduces the concept of a basis in linear algebra and explains how PCA can be viewed as a change of basis. The key idea is to find a new set of basis vectors that best represent the data, with the most important directions corresponding to the largest variances in the data. The tutorial discusses the importance of variance in PCA, noting that directions with the largest variances are likely to contain the most significant information. It also addresses the issue of redundancy in data, where multiple measurements may capture the same information. PCA aims to minimize redundancy by finding a set of orthogonal basis vectors that capture the most variance in the data. The covariance matrix is introduced as a key tool in PCA, capturing the relationships between different variables in the dataset. By diagonalizing the covariance matrix, PCA identifies the principal components, which are the directions of maximum variance. The tutorial explains how to compute these principal components using eigenvector decomposition and singular value decomposition (SVD). The tutorial also discusses the limitations of PCA, noting that it assumes linearity and that the data has a high signal-to-noise ratio. These assumptions may not always hold, and PCA may fail to capture the true structure of the data in such cases. The tutorial concludes by highlighting the importance of understanding the assumptions behind PCA and the need for careful interpretation of the results.This tutorial provides an intuitive and mathematical explanation of Principal Component Analysis (PCA), a widely used technique in data analysis. PCA is a method for reducing the dimensionality of data while preserving as much of the original information as possible. The goal of PCA is to identify the most meaningful basis for re-expressing a dataset, which can help reveal hidden structures and filter out noise. The tutorial begins with a simple example involving the motion of a spring, illustrating how PCA can be used to extract the underlying dynamics from noisy measurements. It then introduces the concept of a basis in linear algebra and explains how PCA can be viewed as a change of basis. The key idea is to find a new set of basis vectors that best represent the data, with the most important directions corresponding to the largest variances in the data. The tutorial discusses the importance of variance in PCA, noting that directions with the largest variances are likely to contain the most significant information. It also addresses the issue of redundancy in data, where multiple measurements may capture the same information. PCA aims to minimize redundancy by finding a set of orthogonal basis vectors that capture the most variance in the data. The covariance matrix is introduced as a key tool in PCA, capturing the relationships between different variables in the dataset. By diagonalizing the covariance matrix, PCA identifies the principal components, which are the directions of maximum variance. The tutorial explains how to compute these principal components using eigenvector decomposition and singular value decomposition (SVD). The tutorial also discusses the limitations of PCA, noting that it assumes linearity and that the data has a high signal-to-noise ratio. These assumptions may not always hold, and PCA may fail to capture the true structure of the data in such cases. The tutorial concludes by highlighting the importance of understanding the assumptions behind PCA and the need for careful interpretation of the results.