Rank (linear algebra)

In linear algebra, the rank of a matrix is the dimension of the vector space generated by its columns. This corresponds to the maximal number of linearly independent columns of. This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics.
The rank is commonly denoted by or ; sometimes the parentheses are not written, as in.

Main definitions

In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these.
The column rank of is the dimension of the column space of, while the row rank of is the dimension of the row space of.
A fundamental result in linear algebra is that the column rank and the row rank are always equal. This number is simply called the rank of.
A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. A matrix is said to be rank-deficient if it does not have full rank. The rank deficiency of a matrix is the difference between the lesser of the number of rows and columns, and the rank.
The rank of a linear map or operator is defined as the dimension of its image:where is the dimension of a vector space, and is the image of a map.

Examples

The matrix
has rank 2: the first two columns are linearly independent, so the rank is at least 2, but since the third is a linear combination of the first two, the three columns are linearly dependent so the rank must be less than 3.
The matrix
has rank 1: there are nonzero columns, so the rank is positive, but any pair of columns is linearly dependent. Similarly, the transpose
of has rank 1. Indeed, since the column vectors of are the row vectors of the transpose of, the statement that the column rank of a matrix equals its row rank is equivalent to the statement that the rank of a matrix is equal to the rank of its transpose, i.e.,.

Computing the rank of a matrix

Rank from row echelon forms

A common approach to finding the rank of a matrix is to reduce it to a simpler form, generally row echelon form, by elementary row operations. Row operations do not change the row space, and, being invertible, map the column space to an isomorphic space. Once in row echelon form, the rank is clearly the same for both row rank and column rank, and equals the number of pivots and also the number of non-zero rows.
For example, the matrix given by
can be put in reduced row-echelon form by using the following elementary row operations:
The final matrix has two non-zero rows and thus the rank of matrix is 2.

Computation

When applied to floating point computations on computers, basic Gaussian elimination can be unreliable, and a rank-revealing decomposition should be used instead. An effective alternative is the singular value decomposition, but there are other less computationally 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.

Proofs that column rank = row rank

Proof using row reduction

The fact that the column and row ranks of any matrix are equal is fundamental in linear algebra. Many proofs have been given. One of the most elementary ones has been sketched in. Here is a variant of this proof:
It is straightforward to show that neither the row rank nor the column rank are changed by an elementary row operation. As Gaussian elimination proceeds by elementary row operations, the reduced row echelon form of a matrix has the same row rank and the same column rank as the original matrix. Further elementary column operations allow putting the matrix in the form of an identity matrix possibly bordered by rows and columns of zeros. Again, this changes neither the row rank nor the column rank. It is immediate that both the row and column ranks of this resulting matrix is the number of its nonzero entries.
We present two other proofs of this result. The first uses only basic properties of linear combinations of vectors, and is valid over any field. The proof is based upon Wardlaw. The second uses orthogonality and is valid for matrices over the real numbers; it is based upon Mackiw. Both proofs can be found in the book by Banerjee and Roy.

Proof using linear combinations

Let be an matrix. Let the column rank of be, and let be any basis for the column space of. Place these as the columns of an matrix. Every column of can be expressed as a linear combination of the columns in. This means that there is an matrix such that. is the matrix whose th column is formed from the coefficients giving the th column of as a linear combination of the columns of. In other words, is the matrix which contains the multiples for the bases of the column space of , which are then used to form as a whole. Now, each row of is given by a linear combination of the rows of. Therefore, the rows of form a spanning set of the row space of and, by the Steinitz exchange lemma, the row rank of cannot exceed. This proves that the row rank of is less than or equal to the column rank of. This result can be applied to any matrix, so apply the result to the transpose of. Since the row rank of the transpose of is the column rank of and the column rank of the transpose of is the row rank of, this establishes the reverse inequality and we obtain the equality of the row rank and the column rank of.

Proof using orthogonality

Let be an matrix with entries in the real numbers whose row rank is. Therefore, the dimension of the row space of is. Let be a basis of the row space of. We claim that the vectors are linearly independent. To see why, consider a linear homogeneous relation involving these vectors with scalar coefficients :
where. We make two observations: is a linear combination of vectors in the row space of, which implies that belongs to the row space of, and since, the vector is orthogonal to every row vector of and, hence, is orthogonal to every vector in the row space of. The facts and together imply that is orthogonal to itself, which proves that or, by the definition of,
But recall that the were chosen as a basis of the row space of and so are linearly independent. This implies that. It follows that are linearly independent.
Every is in the column space of. So, is a set of linearly independent vectors in the column space of and, hence, the dimension of the column space of must be at least as big as. This proves that row rank of is no larger than the column rank of. Now apply this result to the transpose of to get the reverse inequality and conclude as in the previous proof.

Alternative definitions

In all the definitions in this section, the matrix is taken to be an matrix over an arbitrary field.

Dimension of image

Given the matrix, there is an associated linear mapping
defined by
The rank of is the dimension of the image of. This definition has the advantage that it can be applied to any linear map without need for a specific matrix.

Rank in terms of nullity

Given the same linear mapping as above, the rank is minus the dimension of the kernel of. The rank–nullity theorem states that this definition is equivalent to the preceding one.

Column rank – dimension of column space

The rank of is the maximal number of linearly independent columns of ; this is the dimension of the column space of .

Row rank – dimension of row space

The rank of is the maximal number of linearly independent rows of ; this is the dimension of the row space of.

Decomposition rank

The rank of is the smallest positive integer such that can be factored as, where is an matrix and is a matrix. In 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 ,
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 from, take to be the matrix whose columns are from.
To prove from, 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. 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. 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 for the rank of the matrix, which can be useful to prove that certain operations do not lower the rank of a matrix.
A non-vanishing -minor 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, so the row and column rank are at least as large as the determinantal rank; however, 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 : the equivalence implies that a subset of the rows and a subset of the columns simultaneously define an invertible submatrix.

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 if and only if has rank .
is surjective if and only if has rank .
If is a square matrix, then is invertible if and only if has 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 and the three zero matrices have the sizes, and.
Sylvester’s rank inequality: if is an matrix and is, then
The inequality due to Frobenius: if, and are defined, then
Subadditivity:
If is a matrix over the real numbers then the rank of and the rank of its corresponding Gram matrix are equal. Thus, for real matrices This can be shown by proving equality of their null spaces. The null space of the Gram matrix is given by vectors for which If this condition is fulfilled, we also have In fact, since,, which implies and likewise.
If is a matrix over the complex numbers and denotes the complex conjugate of and the conjugate transpose of , then
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. According to the Rouché–Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If on the other hand, the ranks of these two matrices are equal, then the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has free parameters where is the difference between the number of variables and the rank. In this case the system of equations has infinitely many solutions.
In control theory, the rank of a matrix can be used to determine whether a linear system is controllable, or observable.
In the field of communication complexity, the rank of the communication matrix of a function gives bounds on the amount of communication needed for two parties to compute the function.

Generalization

There are different generalizations of the concept of rank to matrices over arbitrary rings, where 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.
Thinking of matrices as tensors, the tensor rank generalizes to arbitrary tensors; for tensors of order greater than 2, rank is very hard to compute, unlike for matrices.
There is a notion of rank for smooth maps between smooth manifolds. It is equal to the linear rank of the derivative.

Matrices as tensors

Matrix rank should not be confused with tensor order, which is called tensor rank. Tensor order is the number of indices required to write a tensor, and thus matrices all have tensor order 2. More precisely, matrices are tensors of type, having one row index and one column index, also called covariant order 1 and contravariant order 1; see Tensor (intrinsic definition) for details.
The tensor rank of a matrix can also mean the minimum number of simple tensors necessary to express the matrix as a linear combination, and that this definition does agree with matrix rank as here discussed.