Rank (linear Algebra) - Alternative Definitions

Alternative Definitions

dimension of image

If one considers the matrix A as a linear mapping f: Fn → Fm such that f(x) = Ax then the rank of A can also be defined as the dimension of the image of f (see linear map for a discussion of image and kernel). This definition has the advantage that it can be applied to any linear map without need for a specific matrix. The rank can also be defined as n minus the dimension of the kernel of f; the rank–nullity theorem states that this is the same as the dimension of the image of f.

column rank – dimension of column space

The maximal number of linearly independent columns of the m × n matrix A with entries in the field F is equal to the dimension of the column space of A (the column space being the subspace of Fm generated by the columns of A, which is in fact just the image of A as a linear map).

row rank – dimension of row space

Since the column rank and the row rank are the same, we can also define the rank of A as the dimension of the row space of A, or the number of rows in a basis of the row space.

decomposition rank

The rank can also be characterized as the decomposition rank: the minimum k such that A can be factored as A = CR, where C is an m×k matrix and R is a k×n matrix. Like the "dimension of image" characterization this can be generalized to a definition of the rank of a linear map: the rank of a linear map f: V → W is the minimal dimension k of an intermediate space X such that f can be written as the composition of a map V → X and a map X → W. While this definition does not suggest an efficient manner to compute the rank (for which it is better to use one of the alternative definitions), it does allow to easily understand many of the properties of the rank, for instance that the rank of the transpose of A is the same as that of A. See rank factorization for details.

determinantal rank – size of largest non-vanishing minor

Another equivalent definition of the rank of a matrix is the greatest order of any non-zero minor in the matrix (the order of a minor being the size 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 to prove that certain operations do not lower the rank of a matrix.

Equivalence of the determinantal definition (rank of largest non-vanishing minor) is generally proved alternatively. It is a generalization of the statement that if the span of n vectors has dimension p, then p of those vectors span the space: one can choose a spanning set that is a subset of the vectors. For determinantal rank, the statement is that if the row rank (column rank) of a matrix is p, then one can choose a p × p submatrix that is invertible: a subset of the rows and a subset of the columns simultaneously define an invertible submatrix. It can be alternatively stated as: if the span of n vectors has dimension p, then p of these vectors span the space and there is a set of p coordinates on which they are linearly independent.

A non-vanishing p-minor (p × p submatrix with non-vanishing 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, the converse is less straightforward.

tensor rank – minimum number of simple tensors

The rank of a square matrix can also be characterized as the tensor rank: the minimum number of simple tensors (rank 1 tensors) needed to express A as a linear combination, . Here a rank 1 tensor (matrix product of a column vector and a row vector) is the same thing as a rank 1 matrix of the given size. This interpretation can be generalized in the separable models interpretation of the singular value decomposition.