In linear algebra, the rank of a matrix is the maximum number of its linearly independent columns, which equals the maximum number of its linearly independent rows. Equivalently, it is the dimension of the image of the linear map represented by the matrix. Rank measures how many independent output directions the matrix can produce, rather than simply how many entries are nonzero. It is usually written (\operatorname{rank}(A)). (arxiv.org)
Definition and interpretation
Let (A) be an (m\times n) matrix over a field (F). Its columns belong to the vector space (F^m), and their span is the column space: [ \operatorname{Col}(A)={Ax:x\in F^n}. ] The rank is its dimension: [ \operatorname{rank}(A)=\dim\operatorname{Col}(A). ] Thus, a rank-(r) matrix has (r) columns forming a basis for its column space; every other column is a linear combination of those basis columns. (math.mit.edu)
The row space is the span of the rows in (F^n). The equality of row rank and column rank is a theorem, not part of the definition of either space. Although these spaces generally lie in different ambient spaces, both have dimension (r). Consequently, the transpose satisfies [ \operatorname{rank}(A^{\mathsf T})=\operatorname{rank}(A). ] (arxiv.org)
Viewed as a linear map (T_A:F^n\to F^m), the matrix sends the domain onto an (r)-dimensional subspace. For example, the real matrix (\operatorname{diag}(1,1,0)) has rank two: it maps three-dimensional space onto the coordinate plane and sends the third coordinate direction to zero. (math.mit.edu)
Equivalent characterizations
Rank always satisfies [ 0\leq\operatorname{rank}(A)\leq\min(m,n). ] The zero matrix has rank zero. A matrix has full column rank when its rank is (n), full row rank when its rank is (m), and full rank when its rank is (\min(m,n)). Otherwise it is rank-deficient. (arxiv.org)
Rank also equals the largest order of a nonzero minor—a determinant obtained by selecting equally many rows and columns. A nonzero (r\times r) minor, together with the vanishing of all larger minors, establishes rank (r). For an (n\times n) matrix, full rank is therefore equivalent to a nonzero determinant and to the existence of an inverse matrix. (math.brown.edu)
Over the real or complex numbers, the singular value decomposition gives [ A=U\Sigma V^, ] where (V^) denotes conjugate transpose. Rank is the number of strictly positive singular values in (\Sigma). These values are the square roots of the positive eigenvalues of (A^*A). (ocw.mit.edu)
Computation by elimination
Gaussian elimination computes rank by reducing a matrix to row-echelon form. Swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another preserve rank. The number of pivot positions, equivalently the number of nonzero rows in echelon form, is the rank. (arxiv.org)
For example, [ A= \begin{pmatrix} 1&2&3\ 2&4&6\ 0&1&1 \end{pmatrix} \quad\longrightarrow\quad \begin{pmatrix} 1&2&3\ 0&1&1\ 0&0&0 \end{pmatrix}. ] Here the second original row is twice the first, while the third is independent of it. There are two pivots, so the rank is two.
To obtain a basis for the original column space, select the original columns corresponding to pivot columns. Row reduction preserves dependencies among columns but need not preserve the column space itself. In this example, the first two original columns form such a basis. (math.mit.edu)
Rank, nullity, and linear systems
The null space is [ \ker A={x\in F^n:Ax=0}. ] Its dimension is called nullity. The rank–nullity theorem states [ \operatorname{rank}(A)+\dim\ker A=n. ] Thus rank counts independent output directions, while nullity counts independent input directions annihilated by the map. (math.mit.edu)
For a system of linear equations (Ax=b), a solution exists exactly when (b) belongs to the column space, equivalently when [ \operatorname{rank}(A)=\operatorname{rank}([A\mid b]). ] When a solution exists, it is unique precisely when (\operatorname{rank}(A)=n). Otherwise the solutions have (n-\operatorname{rank}(A)) free parameters. Over a finite field this still describes their dimension, although the number of solutions is finite. (math.mit.edu)
Numerical rank and low-rank approximation
In numerical linear algebra, exact zero tests are problematic because floating-point arithmetic and measurement uncertainty can turn exact dependencies into small nonzero singular values. Numerical rank therefore counts singular values exceeding a chosen tolerance, rather than every mathematically positive value. The tolerance depends on precision, uncertainty, scaling, and the application; numerical rank is not an intrinsic integer independent of those choices. (netlib.org)
A common relative criterion is [ r_\tau=#{i:\sigma_i>\tau\sigma_1}, ] where (\sigma_1) is the largest singular value and (\tau) is a specified relative threshold. SVD-based least-squares routines use this kind of criterion to determine effective rank. (netlib.org)
Rank also underlies low-rank matrix factorization. Keeping only the (k) largest singular values produces a matrix of rank at most (k); truncated SVD gives a best such approximation in both the spectral and Frobenius norms. This connects rank with dimensionality reduction: an approximately low-rank data matrix can be represented using fewer independent directions while controlling the approximation error. (ocw.mit.edu)