the converse is less straightforward. The equivalence of determinantal rank and column rank is a strengthening of the statement that if the span of vectors has dimension , then of those vectors span the space (equivalently, that one can choose a spanning set that is a subset of the vectors): the equivalence implies that a subset of the rows and a subset of the columns simultaneously define an invertible submatrix (equivalently, if the span of vectors has dimension , then of these vectors span the space and there is a set of coordinates on which they are linearly independent). Tensor rank – minimum number of simple tensors The rank of is the smallest number such that can be written as a sum of rank 1 matrices, where a matrix is defined to have rank 1 if and only if it can be written as a nonzero product of a column vector and a row vector . This notion of rank is called tensor rank; it can be generalized in the separable models interpretation of the singular value decomposition. Properties We assume that is an matrix, and we define the linear map by as above. The rank of an matrix is a nonnegative integer and cannot be greater than either or . That is, A matrix that has rank is said to have full rank; otherwise, the matrix is rank deficient. Only a zero matrix has rank zero. is injective (or "one-to-one") if and only if has rank (in this case, we say that has full column rank). is surjective (or "onto") if and only if has rank (in this case, we say that has full row rank). If is a square matrix (i.e., ), then is invertible if and only if has rank (that is, has full rank). If is any matrix, then If is an matrix of rank , then If is an matrix of rank , then The rank of is equal to if and only if there exists an invertible matrix and an invertible matrix such that where denotes the identity matrix. Sylvester’s rank inequality: if is an matrix and is , then This is a special case of the next inequality. The inequality due to Frobenius: if , and are defined, then Subadditivity: when and are of the same dimension. As a consequence, a rank- matrix can be written as the sum of rank-1 matrices, but