This chapter presents the theory of singular value decomposition (SVD) and principal component analysis (PCA), which are fundamental tools for analyzing and approximating high-dimensional data. We begin by reviewing the Schur decomposition for square matrices and unitary transformations, laying the groundwork for understanding SVD. The chapter then introduces singular value decomposition, proving its existence, uniqueness, and the relationship between singular values and the eigenvalues of \(A^TA\) and \(AA^T\) . Building on SVD, we develop principal component analysis as a method for dimensionality reduction. We define the Rayleigh quotient and formulate a sequence of optimization problems whose solutions yield the principal components of a given matrix. The chapter establishes the orthogonality of left and right singular vectors and demonstrates that the sum of the first k principal components provides the best rank-k approximation of the original matrix in the Frobenius norm.