Singular Value Decomposition (SVD) | Step-By-Step Example and Explanation

Описание: 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

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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.

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