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In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It has many useful applications in signal processing and statistics. Formally, 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 non-negative real numbers on the diagonal, and (the conjugate transpose of , or simply the transpose of if is real) is an real or complex unitary matrix. The diagonal entries of are known as the singular values of . The columns of and the columns of are called the left-singular vectors and right-singular vectors of , respectively. The singular value decomposition and the eigendecomposition are closely related. Namely: : * The left-singular vectors of are eigenvectors of . : * The right-singular vectors of are eigenvectors of . : * The non-zero singular values of (found on the diagonal entries of ) are the square roots of the non-zero eigenvalues of both and . Applications that employ the SVD include computing the pseudoinverse, least squares fitting of data, multivariable control, matrix approximation, and determining the rank, range and null space of a matrix. == Statement of the theorem == Suppose is a matrix whose entries come from the field , which is either the field of real numbers or the field of complex numbers. Then there exists a factorization, called a singular value decomposition of , of the form : where * is a , unitary matrix, * is a diagonal matrix with non-negative real numbers on the diagonal, and * is a , unitary matrix over . (If , unitary matrices are orthogonal matrices.) is the conjugate transpose of the unitary matrix, . The diagonal entries, , of are known as the singular values of . A common convention is to list the singular values in descending order. In this case, the diagonal matrix, , is uniquely determined by (though not the matrices and ). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「singular value decomposition」の詳細全文を読む スポンサード リンク
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