Editing Rank (linear algebra) (section)

== Properties ==
We assume that {{mvar|A}} is an {{math|''m'' × ''n''}} matrix, and we define the linear map {{mvar|f}} by {{math|1=''f''('''x''') = ''A'''''x'''}} as above.

* The rank of an {{math|1=''m'' × ''n''}} matrix is a [[nonnegative]] [[integer]] and cannot be greater than either {{mvar|m}} or {{mvar|n}}. That is, <math display="block">\operatorname{rank}(A) \le \min(m, n).</math> A matrix that has rank {{math|min(''m'', ''n'')}} is said to have ''full rank''; otherwise, the matrix is ''rank deficient''.
* Only a [[zero matrix]] has rank zero.
* {{mvar|f}} is [[injective function|injective]] (or "one-to-one") if and only if {{mvar|A}} has rank {{mvar|n}} (in this case, we say that {{mvar|A}} has ''full column rank'').
* {{mvar|f}} is [[surjective function|surjective]] (or "onto") if and only if {{mvar|A}} has rank {{mvar|m}} (in this case, we say that {{mvar|A}} has ''full row rank'').
* If {{mvar|A}} is a square matrix (i.e., {{math|1=''m'' = ''n''}}), then {{mvar|A}} is [[invertible matrix|invertible]] if and only if {{mvar|A}} has rank {{mvar|n}} (that is, {{mvar|A}} has full rank).
* If {{mvar|B}} is any {{math|''n'' × ''k''}} matrix, then <math display="block">\operatorname{rank}(AB) \leq \min(\operatorname{rank}(A), \operatorname{rank}(B)).</math>
* If {{mvar|B}} is an {{math|''n'' × ''k''}} matrix of rank {{mvar|n}}, then <math display="block">\operatorname{rank}(AB) = \operatorname{rank}(A).</math>
* If {{mvar|C}} is an {{math|''l'' × ''m''}} matrix of rank {{mvar|m}}, then <math display="block">\operatorname{rank}(CA) = \operatorname{rank}(A).</math>
* The rank of {{mvar|A}} is equal to {{mvar|r}} if and only if there exists an invertible {{math|''m'' × ''m''}} matrix {{mvar|X}} and an invertible {{math|''n'' × ''n''}} matrix {{mvar|Y}} such that <math display="block"> XAY =
\begin{bmatrix}
  I_r & 0 \\
  0 & 0
\end{bmatrix},</math> where {{math|''I''<sub>''r''</sub>}} denotes the {{math|''r'' × ''r''}} [[identity matrix]] and the three [[zero matrix|zero matrices]] have the sizes {{math|''r'' × (''n'' − ''r'')}}, {{math|(''m'' − ''r'') × ''r''}} and {{math|(''m'' − ''r'') × (''n'' − ''r'')}}.
* '''[[James Joseph Sylvester|Sylvester]]’s rank inequality''': if {{mvar|A}} is an {{math|''m'' × ''n''}} matrix and {{mvar|B}} is {{math|''n'' × ''k''}}, then<ref group="lower-roman">Proof: Apply the [[rank–nullity theorem]] to the inequality <math display="block">\dim \ker(AB) \le \dim \ker(A) + \dim \ker(B).</math></ref> <math display="block">\operatorname{rank}(A) + \operatorname{rank}(B) - n \leq \operatorname{rank}(A B).</math> This is a special case of the next inequality.
* The inequality due to [[Ferdinand Georg Frobenius|Frobenius]]: if {{math|''AB''}}, {{math|''ABC''}} and {{math|''BC''}} are defined, then<ref group="lower-roman">Proof. The map<math display="block">C: \ker(ABC) / \ker(BC) \to \ker(AB) / \ker(B)</math>is well-defined and injective. We thus obtain the inequality in terms of dimensions of kernel, which can then be converted to the inequality in terms of ranks by the [[rank–nullity theorem]].
Alternatively, if <math>M</math> is a linear subspace then <math>\dim (AM) \leq \dim (M)</math>; apply this inequality to the subspace defined by the orthogonal complement of the image of <math>BC</math> in the image of <math>B</math>, whose dimension is <math>\operatorname{rank} (B) - \operatorname{rank} (BC)</math>; its image under <math>A</math> has dimension <math>\operatorname{rank} (AB) - \operatorname{rank} (ABC)</math>.</ref> <math display="block">\operatorname{rank}(AB) + \operatorname{rank}(BC) \le \operatorname{rank}(B) + \operatorname{rank}(ABC).</math>
* Subadditivity: <math display="block">\operatorname{rank}(A+ B) \le \operatorname{rank}(A) + \operatorname{rank}(B) </math> when {{mvar|A}} and {{mvar|B}} are of the same dimension.  As a consequence, a rank-{{mvar|k}} matrix can be written as the sum of {{mvar|k}} rank-1 matrices, but not fewer.
* The rank of a matrix plus the [[Kernel (matrix)|nullity]] of the matrix equals the number of columns of the matrix. (This is the [[rank–nullity theorem]].)
* If {{mvar|A}} is a matrix over the [[real numbers]] then the rank of {{mvar|A}} and the rank of its corresponding [[Gram matrix]] are equal. Thus, for real matrices <math display="block">\operatorname{rank}(A^\mathrm{T} A) = \operatorname{rank}(A A^\mathrm{T}) = \operatorname{rank}(A) = \operatorname{rank}(A^\mathrm{T}).</math> This can be shown by proving equality of their [[kernel (matrix)|null spaces]]. The null space of the Gram matrix is given by vectors {{math|'''x'''}} for which <math>A^\mathrm{T} A \mathbf{x} = 0.</math> If this condition is fulfilled, we also have <math>0 = \mathbf{x}^\mathrm{T} A^\mathrm{T} A \mathbf{x} = \left| A \mathbf{x} \right| ^2.</math><ref>{{cite book| last = Mirsky| first = Leonid| title = An introduction to linear algebra| year = 1955| publisher = Dover Publications| isbn = 978-0-486-66434-7 }}</ref>
* If {{mvar|A}} is a matrix over the [[complex numbers]] and <math>\overline{A}</math> denotes the complex conjugate of {{mvar|A}} and {{math|''A''<sup>∗</sup>}} the conjugate transpose of {{mvar|A}} (i.e., the [[Hermitian adjoint|adjoint]] of {{mvar|A}}), then <math display="block">\operatorname{rank}(A) = \operatorname{rank}(\overline{A}) = \operatorname{rank}(A^\mathrm{T}) = \operatorname{rank}(A^*) = \operatorname{rank}(A^*A) = \operatorname{rank}(AA^*).</math>