rdfs:comment
  In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an real or complex unitary matrix. If is real, and are real orthonormal matrices.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an real or complex unitary matrix. If is real, and are real orthogonal matrices.
 In linear algebra, the singsssssssssssssssssular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an real or complex unitary matrix. If is real, and are real orthogonal matrices.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an complex matrix is a factorization of the form , where is an complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an complex unitary matrix. If is real, and are real orthogonal matrices.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of 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 nonnegative real numbers on the diagonal, and V is an complex unitary matrix. If M is real, U and are real orthogonal matrices.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of 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 nonnegative real numbers on the diagonal, and V is an complex unitary matrix. If M is real, U and V are real orthogonal matrices.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. The diagonal entries of are known as the singular values of M. The number of nonzero singular values is equal to the rank of M. The columns of U and the columns of V are called the leftsingular vectors and rightsingular vectors of M, respectively.

has abstract
  In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an real or complex unitary matrix. If is real, and are real orthonormal matrices. The diagonal entries of are known as the singular values of . The number of nonzero singular values is equal to the rank of . The columns of and the columns of are called the leftsingular vectors and rightsingular vectors of , respectively. The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, Σ (but not always U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition in which Σ is square diagonal of size , where is the rank of M, and has only the nonzero singular values. In this variant, is an matrix and is an matrix, such that . 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.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an real or complex unitary matrix. If is real, and are real orthogonal matrices. The diagonal entries of are known as the singular values of . The number of nonzero singular values is equal to the rank of . The columns of and the columns of are called the leftsingular vectors and rightsingular vectors of , respectively. The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, Σ (but not always U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition in which Σ is square diagonal of size , where is the rank of M, and has only the nonzero singular values. In this variant, is an matrix and is an matrix, such that . 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.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an real or complex unitary matrix. If is real, and are real orthogonal matrices. The diagonal entries of are known as the singular values of . The number of nonzero singular values is equal to the rank of . The columns of and the columns of are called the leftsingular vectors and rightsingular vectors of , respectively. The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, (but not always U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition in which is square diagonal of size , where is the rank of M, and has only the nonzero singular values. In this variant, is an matrix and is an matrix, such that . 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.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an real or complex unitary matrix. If is real, and are real orthogonal matrices. The diagonal entries of are known as the singular values of . The number of nonzero singular values is equal to the rank of . The columns of and the columns of are called the leftsingular vectors and rightsingular vectors of , respectively. The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, (but not always U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition in which is square diagonal of size , where is the rank of M, and has only the nonzero singular values. In this variant, is an semiunitary matrix and is an semiunitary matrix, such that . 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.
 In linear algebra, the singsssssssssssssssssular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an real or complex unitary matrix. If is real, and are real orthogonal matrices. The diagonal entries of are known as the singular values of . The number of nonzero singular values is equal to the rank of . The columns of and the columns of are called the leftsingular vectors and rightsingular vectors of , respectively. The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, (but not always U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition in which is square diagonal of size , where is the rank of M, and has only the nonzero singular values. In this variant, is an semiunitary matrix and is an semiunitary matrix, such that . 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.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of the polar decomposition. Specifically, the singular value decomposition of an complex matrix is a factorization of the form , where is an complex unitary matrix, is an rectangular diagonal matrix with nonnegative real numbers on the diagonal, and is an complex unitary matrix. If is real, and are real orthogonal matrices. The diagonal entries of are known as the singular values of . The number of nonzero singular values is equal to the rank of . The columns of and the columns of are called the leftsingular vectors and rightsingular vectors of , respectively. The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, (but not always U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition in which is square diagonal of size , where is the rank of M, and has only the nonzero singular values. In this variant, is an semiunitary matrix and is an semiunitary matrix, such that . 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.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of 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 nonnegative real numbers on the diagonal, and V is an complex unitary matrix. If M is real, U and are real orthogonal matrices. The diagonal entries of are known as the singular values of M. The number of nonzero singular values is equal to the rank of M. The columns of U and the columns of V are called the leftsingular vectors and rightsingular vectors of M, respectively. The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, (but not always U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition in which is square diagonal of size , where is the rank of M, and has only the nonzero singular values. In this variant, U is an semiunitary matrix and is an semiunitary matrix, such that . 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.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of 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 nonnegative real numbers on the diagonal, and V is an complex unitary matrix. If M is real, U and V are real orthogonal matrices. The diagonal entries of are known as the singular values of M. The number of nonzero singular values is equal to the rank of M. The columns of U and the columns of V are called the leftsingular vectors and rightsingular vectors of M, respectively. The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, (but not always U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition in which is square diagonal of size , where is the rank of M, and has only the nonzero singular values. In this variant, U is an semiunitary matrix and is an semiunitary matrix, such that . 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.
 In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix that generalizes the eigendecomposition of a square normal matrix to any matrix via an extension of 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 nonnegative real numbers on the diagonal, and V is an complex unitary matrix. If M is real, factorizations where U and V are real orthogonal matrices exist, and they are the only valid factorizations in contexts where all variables are assumed to be real. In such contexts, the SVD is often denoted . The diagonal entries of are known as the singular values of M. The number of nonzero singular values is equal to the rank of M. The columns of U and the columns of V are called the leftsingular vectors and rightsingular vectors of M, respectively. The SVD is not unique. It is always possible to choose the decomposition so that the singular values are in descending order. In this case, (but not always U and V) is uniquely determined by M. The term sometimes refers to the compact SVD, a similar decomposition in which is square diagonal of size , where is the rank of M, and has only the nonzero singular values. In this variant, U is an semiunitary matrix and is an semiunitary matrix, such that . 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.
