The Singular Value Decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square matrix. This factorization provides a useful representation of the matrix in terms of its singular values and singular vectors.
The Wikipedia page on Singular Value Decomposition provides a comprehensive introduction to the topic. It explains that the SVD factorizes a matrix A into three matrices: U, Σ, and V. The matrix U contains the left singular vectors, Σ is a diagonal matrix with the singular values, and V contains the right singular vectors. This factorization has various applications in linear algebra, signal processing, and data analysis.
The MIT webpage on Singular Value Decomposition provides a tutorial on SVD. It explains that the columns of U represent the left singular vectors, which are also known as gene coefficient vectors. The matrix S is a diagonal matrix that contains the singular values of A. The tutorial also provides examples and code snippets to demonstrate the computation of the SVD.
The GeeksforGeeks page on Singular Value Decomposition goes into more detail about the algebraic properties of SVD. It explains that the SVD factorization can be viewed as a way to decompose a matrix A into the product of three matrices: U, Σ, and V. The matrices U and V are orthogonal matrices, and Σ is a diagonal matrix containing the singular values of A. The page provides examples and explanations of how to compute the SVD using various methods.
A YouTube video titled “Singular Value Decomposition (SVD): Overview” gives a concise overview of the topic. The video explains that the SVD is widely used in various fields for data analysis and dimensionality reduction. It provides an intuitive explanation of how the SVD factors a matrix into its component parts.
An article on Towards Data Science titled “Singular Value Decomposition (SVD), Demystified” provides a more in-depth explanation of SVD. It describes the SVD as a powerful matrix factorization technique that decomposes a matrix into three other matrices. The article explains the mathematical concepts behind SVD and provides examples to illustrate its applications in machine learning and data analysis.
In summary, the Singular Value Decomposition (SVD) is a factorization of a matrix into three matrices: U, Σ, and V. It has applications in linear algebra, signal processing, and data analysis. The SVD can be computed using various methods and is widely used for data analysis and dimensionality reduction.