Lecture

Singular Value Decomposition

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Description

This lecture covers the Singular Value Decomposition (SVD) theorem, which states that for a matrix A of rank r, there exists a matrix U and V forming an orthogonal base of Rn, and a diagonal matrix Σ with singular values. The lecture explains how to factorize A as A = UΣV^T, where Σ is a diagonal matrix with singular values. It also discusses the properties of singular values and their relation to eigenvectors, emphasizing the orthogonality of the base formed by the eigenvectors of ATA. The lecture concludes with a proof of the SVD theorem and its application in decomposing matrices.

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Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.

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LU decomposition

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.

Singular value decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an complex matrix M is a factorization of the form where U is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an complex unitary matrix, and is the conjugate transpose of V.

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