Intuition behind the Singular Value Decomposition SVD

In this video, we will learn how to prove the singular value decomposition (SVD), a powerful factorization of any matrix into three matrices that reveal its structure and properties. For the sake of convenience, we will restrict ourselves to real matrices. The SVD states that any m×n matrix A can be written in the form: A = UΣV', where U is an m×m orthogonal matrix, Σ is an m×n diagonal matrix with non-negative entries, and V is an n×n orthogonal matrix (V' is the transpose of V).
The SVD can help us simplify computations, such as finding inverses, ranks, norms, or determinants of matrices, solve systems of linear equations or least squares problems, and understand the geometric meaning and properties of matrices.
We will show how to prove the existence and uniqueness of the SVD using eigenvalue decomposition and Gram matrices. We will explain how to find the eigenvalues and eigenvectors of A'A and AA' , which are symmetric and positive semi-definite matrices, and how to relate them to the singular values and singular vectors of A. We will also explain how to construct U and V from the eigenvectors of A'A and AA' , respectively, and how to ensure that they are orthogonal matrices. We will also show that Σ is uniquely determined by the singular values of A.
By the end of this lecture, you should be able to:
Recognize and write down the definition of the SVD and its components.
Prove the existence and uniqueness of the SVD using eigenvalue decomposition and Gram matrices.
Construct U, Σ, and V from the eigenvalues and eigenvectors of A'A and AA'.
Apply the SVD to various problems and examples.
By the end of this lecture, you should be able to state and prove the diagonalization theorem.
This video is an excerpt from the course titled "Advanced Data Analysis using Wavelets and Machine Learning".

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