Rank (linear algebra)

In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent.

The column rank and the row rank are equal; this common number is simply called the rank of A. It is commonly denoted by either rk(A) or rank A.

Alternative definitions

The maximal number of linearly independent columns of the m-by-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 F^m generated by the columns of A). 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.

If one considers the matrix A as a linear map

f : Fⁿ → F^m

with the rule

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 they 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.

Another equivalent definition of the rank of a matrix is the order of the greatest non-vanishing minor in the matrix.

Properties

We assume that A is an m-by-n matrix over the field F and describes a linear map f as above.

only the zero matrix has rank 0
the rank of A is at most min(m,n)
f is injective if and only if A has rank n (in this case, we say that A has full column rank).
f is surjective if and only if A has rank m (in this case, we say that A has full row rank).
In the case of a square matrix A (i.e., m = n), then A is invertible if and only if A has rank n (we say that A has full rank).
If B is any n-by-k matrix, then the rank of AB is at most the minimum of the rank of A and the rank of B.

As an example of the "<" case, consider the product

\begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ \end{bmatrix}

Both factors have rank 1, but the product has rank 0.

If B is an n-by-k matrix with rank n, then AB has the same rank as A.
If C is an l-by-m matrix with rank m, then CA has the same rank as A.
The rank of A is equal to r if and only if there exists an invertible m-by-m matrix X and an invertible n-by-n matrix Y such that

XAY = \begin{bmatrix} I_r & 0 \\ 0 & 0 \\ \end{bmatrix}

where I_r denotes the r-by-r identity matrix.

The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix (this is the "rank theorem" or the "rank-nullity theorem").

Computation

The easiest way to compute the rank of a matrix A is given by the Gauss elimination method. The row-echelon form of A produced by the Gauss algorithm has the same rank as A, and its rank can be read off as the number of non-zero rows.

Consider for example the 4-by-4 matrix

A = \begin{bmatrix} 2 & 4 & 1 & 3 \\ -1 & -2 & 1 & 0 \\ 0 & 0 & 2 & 2 \\ 3 & 6 & 2 & 5 \\ \end{bmatrix}

We see that the second column is twice the first column, and that the fourth column equals the sum of the first and the third. The first and the third columns are linearly independent, so the rank of A is two. This can be confirmed with the Gauss algorithm. It produces the following row echelon form of A:

A = \begin{bmatrix} 1 & 2 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}

which has two non-zero rows.

When applied to floating point computations on computers, basic Gaussian elimination (LU decomposition) can be unreliable, and a rank revealing decomposition should be used instead. An effective alternative is the singular value decomposition (SVD), but there are other less expensive choices, such as QR decomposition with pivoting, which are still more numerically robust than Gaussian elimination. Numerical determination of rank requires a criterion for deciding when a value, such as a singular value from the SVD, should be treated as zero, a practical choice which depends on both the matrix and the application.

Applications

One useful application of calculating the rank of a matrix is the computation of the number of solutions of a system of linear equations. The system has at least one solution if the rank of the coefficient matrix equals the rank of augmented matrix. In that case, it has precisely one solution if the rank equals the number of equations; otherwise, the general solution has k free parameters where k is the difference between the number of equations and the rank.

In control theory, the rank of a matrix can be used to determine whether a linear system is controllable, or observable.

Generalization

There are different generalisations of the concept of rank to matrices over arbitrary rings. In those generalisations, column rank, row rank, dimension of column space and dimension of row space of a matrix may be different from the others or may not exist.

References

Horn, Roger A. and Johnson, Charles R. Matrix Analysis. Cambridge University Press, 1985. ISBN 0-521-38632-2.