"In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix."
Singular value decomposition is a factorization of a matrix that expresses it as the product of three matrices. It is useful for many applications, including image compression, data mining, and regression analysis.
Matrices and Vectors: Understanding the concept of matrices and vectors in linear algebra, including their properties and operations.
Linear Transformations: Understanding how matrices represent linear transformations, including rotation, scaling or reflections in higher dimensions.
Eigenvectors and Eigenvalues: Understanding the concept of eigenvectors and eigenvalues, which play crucial roles in many mathematical applications, including linear algebra, statistics, and scientific computing.
Matrix Factorization: Understanding how matrices can be factored into two or more smaller matrices, including the SVD decomposition, QR factorization, and LU decomposition.
Orthogonal Matrices: Understanding the concept of orthogonal matrices, which play an important role in the SVD decomposition and many other linear algebra applications.
Singular Value Decomposition: Understanding how the Singular Value Decomposition (SVD) decomposes any matrix into a product of three matrices, including the left and right singular vectors and the singular values.
Applications of SVD: Understanding the various applications of SVD in different fields, including image compression, signal processing, data analysis, and machine learning.
Dimensionality Reduction: Understanding how SVD can be used for dimensionality reduction, where large datasets can be compressed into a lower dimensional space, while preserving most of their important features.
Truncated SVD: Understanding how truncated SVD can be used for approximating a matrix using a smaller number of singular values and vectors.
Non-negative Matrix Factorization: Understanding how Non-negative matrix factorization (NMF) is a matrix decomposition method that factorizes a matrix into two matrices, and is often used for feature extraction and clustering.
Full SVD: This type of SVD is the most common and provides all the three matrices- U, Sigma, and V. It is used for rank computation, principal component analysis, and noise reduction.
Compact SVD: As the name suggests, it is a more concise version of full-SVD that eliminates the zero rows in the Sigma matrix. This eliminates the non-zero entries of U and V. Thus Compact SVD is used to save space in memory and computation and is used for image and audio compression.
Truncated SVD: This type of SVD keeps only a few significant singular values by zeroing out lower singular values in the Sigma matrix. It is used to reduce noise, data compression, summarization, and dimensionality reduction.
Thin SVD: Thin SVD is similar to the Compact version but also has zero columns removed from the U matrix. It gives a more reduced version of SVD and is applied more in latent semantic analysis and text mining applications.
Eckart-Young-Mirsky Theorem: This theorem states that a matrix can be approximated by SVD such that the error is minimized with the smallest possible rank. This theorem provides a practical approach for dimensionality reduction in applications such as facial recognition, DNA analysis, and collaborative filtering.
Blocked SVD: This type of SVD is used for very large matrices passed as blocks or matrix partitions.
Randomized SVD: This is a matrix approximation algorithm that randomly selects smaller subsets of available data to compute a truncated SVD matrix close to the actual SVD. It is used in big data applications and time-series prediction.
Polar SVD: It establishes the decomposition of a matrix using polar form. It is used in robotics, signal processing, and computational geometry.
Generalized SVD: This type of SVD is used when solving matrix equations that arise from a least-square solution for overdetermined or underdetermined systems. It is used in multivariate statistics and control systems.
Rank-revealing SVD: It is a form of truncated SVD that uncovers and displays the rank of the original matrix. This is useful in data analysis, image processing, and machine learning.
Partial SVD: This is a variation of the ordinary SVD that allows you to obtain the k leading singular values and eigenvectors, along with the nonzero singular values that occurred. It is used in quantum mechanics and quantum cryptography.
"It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix."
"M = UΣV*, where U is an m×m complex unitary matrix, Σ is an m×n rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an n×n complex unitary matrix, and V* is the conjugate transpose of V."
"The diagonal entries σi of Σ are uniquely determined by M and are known as the singular values of M."
"The number of non-zero singular values is equal to the rank of M."
"The columns of U and the columns of V are called left-singular vectors and right-singular vectors of M, respectively."
"It is related to the polar decomposition."
"They form two sets of orthonormal bases u1, ..., um and v1, ..., vn."
"M = ∑i=1r σi ui vi*, where r≤min{m,n} is the rank of M."
"The SVD is not unique."
"It is always possible to choose the decomposition so that the singular values Σii are in descending order."
"The term sometimes refers to the compact SVD, a similar decomposition."
"Σ is a square diagonal of size r×r, where r≤min{m,n} is the rank of M."
"In this variant, U is an m×r semi-unitary matrix and V is an n×r semi-unitary matrix."
"Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix."
"The SVD is also extremely useful in all areas of science, engineering, and statistics, such as signal processing, least squares fitting of data, and process control." Quotes providing answers: