= Singular Values = '''Singular values''' are a generalization of [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. <> ---- == Introduction == It is known that (for square matrices) there are [[LinearAlgebra/EigenvaluesAndEigenvectors|certain vectors]] which the transformation only scales: '''''A'''x = λx''. Similarly, for any matrix '''''A''''' of size ''m x n'', there are certain vectors ''u,,i,,'' in the column space that correspond to certain vectors ''v,,i,,'' in the row space. They correspond such that '''''A'''v = σu''. And by composing these vectors and values together: '''''AV''' = '''UΣ'''''. The size of '''''Σ''''' will always match '''''A'''''; that is, it will also be size ''m x n''. '''''V''''' spans the column space of '''''A''''', so it will be size ''n x n''. '''''U''''' spans the row space, so it will be ''m x m''. ---- == Description == Singular values are square roots of [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. Specifically, a matrix '''''A''''' has singular values equal to the square roots of the eigenvalues of '''''A'''^T^'''A''''': ''σ = √λ''. Matrices of all sizes have singular values. The number of non-zero singular values is equal to the [[LinearAlgebra/Rank|rank]] of '''''A'''''. Singular values are also more stable as compared to eigenvalues. Consider the matrix: {{{ ┌ ┐ │ 0 1 0 0│ │ 0 0 2 0│ │ 0 0 0 3│ │ 0 0 0 0│ └ ┘ }}} All eigenvalues are 0 and there is only one eigenvector: ''[1 0 0 0]''. By comparison, the singular values are 3, 2, 1, and 0. If the 0 in row 4, column 1 were changed to an extremely small degree, then 4 unique eigenpairs would suddenly emerge. The first three singular values would be unchanged however, and the fourth would become non-zero. ---- CategoryRicottone