Learn Singular Value Decomposition (SVD) in this step-by-step tutorial, where we break down the process using a detailed example. This video explains how to find the SVD of a matrix (A=UΣV^T), covering key steps such as constructing Σ, U, and V, and understanding their structure and geometrical significance in linear algebra. We’ll guide you through: ✅ Finding the sizes of Σ, U, and V, and the rank of A ✅ Constructing Σ by finding the eigenvalues of (A^T)A and singular values of A ✅ Constructing V by finding the normalized eigenvectors of (A^T)A and the additional normalized vectors from Ax=0. ✅ Constructing U from the vectors in V and finding the additional normalized vectors from A^Tx=0 ✅ Constructing and verifying the decomposition A=UΣV^T *Link to Guided Notes* Free downloadable pdf of guided notes that will help you follow along with the video and use later to review what you learned. _Linear Equations in Linear Algebra:_ _Matrix Algebra:_ _Determinants_ _Vector Spaces_ _Eigenvalues and Eigenvectors_ _Orthogonality and Least Squares_ *Subscribe to My YouTube Channel* *Visit the Understand The Math Website* Find free guided notes, filled in guided notes for purchase, course outlines/playlists, and math merchandise. *Connect on Social Media* Instagram: Facebook: Pinterest: *Timestamps* 00:00 Introduction 00:25 SVD Structure 04:50 Step 1 - Find matrix sizes and rank of A 06:14 Step 2 - Find matrix Sigma 09:00 Step 3 - Find matrix V 12:54 Step 4 - Find matrix U 17:34 Step 5 - Construct SVD *More About Understand The Math* Understand The Math offers clear, accessible, and engaging mathematics instruction for students and lifelong learners. I'm Cheryl Hile, the founder of Understand The Math, with a Ph.D. in Engineering Science and Applied Mathematics and over 25 years of university teaching experience. My goal is to provide accurate, high-quality math instruction through step-by-step explanations and carefully worked-out examples. Each video includes free guided notes, linked in the description. These notes help students follow along, highlight key concepts, and practice problems, making review easier and more effective. #LinearAlgebra #SingularValueDecomposition #SVD #SVDExample #MatrixDecomposition #Eigenvalues #Eigenvectors #MathTutorial #EngineeringMath #MathHelp #UniversityMath #AdvancedMathematics #MatrixAlgebra