Editing Rank (linear algebra) (section)

== Alternative definitions ==
In all the definitions in this section, the matrix {{mvar|A}} is taken to be an {{math|''m'' × ''n''}} matrix over an arbitrary [[Field (mathematics)|field]] {{mvar|F}}.

=== Dimension of image ===
Given the matrix <math>A</math>, there is an associated [[linear mapping]]
<math display="block">f : F^n \to F^m</math>
defined by
<math display="block">f(x) = Ax.</math>
The rank of <math>A</math> is the dimension of the image of <math>f</math>.  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 {{mvar|f}} as above, the rank is {{mvar|n}} minus the dimension of the [[kernel (algebra)|kernel]] of {{mvar|f}}.  The [[rank–nullity theorem]] states that this definition is equivalent to the preceding one.

=== Column rank – dimension of column space ===
The rank of {{mvar|A}} is the maximal number of linearly independent columns <math>\mathbf{c}_1,\mathbf{c}_2,\dots,\mathbf{c}_k</math> of {{mvar|A}}; this is the [[dimension of a vector space|dimension]] of the [[column space]] of {{mvar|A}} (the column space being the subspace of {{math|''F''<sup>''m''</sup>}} generated by the columns of {{mvar|A}}, which is in fact just the image of the linear map {{mvar|f}} associated to {{mvar|A}}).

=== Row rank – dimension of row space ===
The rank of {{mvar|A}} is the maximal number of linearly independent rows of {{mvar|A}}; this is the dimension of the [[row space]] of {{mvar|A}}.

=== Decomposition rank ===
The rank of {{mvar|A}} is the smallest positive integer {{mvar|k}} such that {{mvar|A}} can be factored as <math>A = CR</math>, where {{mvar|C}} is an {{math|''m'' × ''k''}} matrix and {{mvar|R}} is a {{math|''k'' × ''n''}} matrix. In fact, for all integers {{mvar|k}}, the following are equivalent:

# the column rank of {{mvar|A}} is less than or equal to {{mvar|k}},
# there exist {{mvar|k}} columns <math>\mathbf{c}_1,\ldots,\mathbf{c}_k</math> of size {{mvar|m}} such that every column of {{mvar|A}} is a linear combination of <math>\mathbf{c}_1,\ldots,\mathbf{c}_k</math>,
# there exist an <math>m \times k</math> matrix {{mvar|C}} and a <math>k \times n</math> matrix {{mvar|R}} such that <math>A = CR</math> (when {{mvar|k}} is the rank, this is a [[rank factorization]] of {{mvar|A}}),
# there exist {{mvar|k}} rows <math>\mathbf{r}_1,\ldots,\mathbf{r}_k</math> of size {{mvar|n}} such that every row of {{mvar|A}} is a linear combination of <math>\mathbf{r}_1,\ldots,\mathbf{r}_k</math>,
# the row rank of {{mvar|A}} is less than or equal to {{mvar|k}}.

Indeed, the following equivalences are obvious: <math>(1)\Leftrightarrow(2)\Leftrightarrow(3)\Leftrightarrow(4)\Leftrightarrow(5)</math>.
For example, to prove (3) from (2), take {{mvar|C}} to be the matrix whose columns are <math>\mathbf{c}_1,\ldots,\mathbf{c}_k</math> from (2).
To prove (2) from (3), take <math>\mathbf{c}_1,\ldots,\mathbf{c}_k</math> to be the columns of {{mvar|C}}.

It follows from the equivalence <math>(1)\Leftrightarrow(5)</math> 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 {{math|''f'' : ''V'' → ''W''}} is the minimal dimension {{mvar|k}} of an intermediate space {{mvar|X}} such that {{mvar|f}} can be written as the composition of a map {{math|''V'' → ''X''}} and a map {{math|''X'' → ''W''}}. Unfortunately, this definition does not suggest an efficient manner to compute the rank (for which it is better to use one of the alternative definitions). See [[rank factorization]] for details.

=== Rank in terms of singular values ===
The rank of {{mvar|A}} equals the number of non-zero [[Singular value decomposition|singular values]], which is the same as the number of non-zero diagonal elements in Σ in the [[singular value decomposition]] {{nowrap|<math>A = U \Sigma V^*</math>.}}

=== Determinantal rank – size of largest non-vanishing minor ===
The rank of {{mvar|A}} is the largest order of any non-zero [[Minor (linear algebra)|minor]] in {{mvar|A}}.  (The order of a minor is the side-length 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 (for example) to prove that certain operations do not lower the rank of a matrix.

A non-vanishing {{mvar|p}}-minor ({{math|''p'' × ''p''}} submatrix with non-zero 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.  The equivalence of determinantal rank and column rank is a strengthening of the statement that if the span of {{mvar|n}} vectors has dimension {{mvar|p}}, then {{mvar|p}} 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 {{mvar|n}} vectors has dimension {{mvar|p}}, then {{mvar|p}} of these vectors span the space ''and'' there is a set of {{mvar|p}} coordinates on which they are linearly independent).

=== Tensor rank – minimum number of simple tensors ===
{{Main|Tensor rank decomposition|Tensor rank}}

The rank of {{mvar|A}} is the smallest number {{mvar|k}} such that {{mvar|A}} can be written as a sum of {{mvar|k}} rank 1 matrices, where a matrix is defined to have rank 1 if and only if it can be written as a nonzero product <math>c \cdot r</math> of a column vector {{mvar|c}} and a row vector {{mvar|r}}.  This notion of rank is called [[tensor rank]]; it can be generalized in the [[Singular value decomposition#Separable models|separable models]] interpretation of the [[singular value decomposition]].