fact, for all integers , the following are equivalent: the column rank of is less than or equal to , there exist columns of size such that every column of is a linear combination of , there exist an matrix and a matrix such that (when is the rank, this is a rank factorization of ), there exist rows of size such that every row of is a linear combination of , the row rank of is less than or equal to . Indeed, the following equivalences are obvious: . For example, to prove (3) from (2), take to be the matrix whose columns are from (2). To prove (2) from (3), take to be the columns of . It follows from the equivalence that the row rank is equal to the column rank. As in the case of the "dimension of image" characterization, this can be generalized to a definition of the rank of any linear map: the rank of a linear map is the minimal dimension of an intermediate space such that can be written as the composition of a map and a map . Unfortunately, this definition does not suggest an efficient manner to compute the rank (for which it is better to use one of the alternative definitions). See rank factorization for details. Rank in terms of singular values The rank of equals the number of non-zero singular values, which is the same as the number of non-zero diagonal elements in Σ in the singular value decomposition Determinantal rank – size of largest non-vanishing minor The rank of is the largest order of any non-zero minor in . (The order of a minor is the side-length of the square sub-matrix of which it is the determinant.) Like the decomposition rank characterization, this does not give an efficient way of computing the rank, but it is useful theoretically: a single non-zero minor witnesses a lower bound (namely its order) for the rank of the matrix, which can be useful (for example) to prove that certain operations do not lower the rank of a matrix. A non-vanishing -minor ( submatrix with non-zero determinant) shows that the rows and columns of that submatrix are linearly independent, and thus those rows and columns of the full matrix are linearly independent (in the full matrix), so the row and column rank are at least as large as the determinantal rank; however,