Rank (math)

The title of this article is ambiguous. The rank of a matrix is ​​dealt with here, for the rank of an Abelian group see Rank of an Abelian group , for that of a free group see Free Group # Rank .

The ranking is a term used in linear algebra . You assign it to a matrix or a linear mapping . Common spellings are and . The English spellings and are also used less frequently. ${\displaystyle \mathrm {rang} (f)}$ ${\displaystyle \mathrm {rg} (f)}$ ${\displaystyle \mathrm {rank} (f)}$ ${\displaystyle \mathrm {rk} (f)}$

definition

Column vectors of a matrix

• For a matrix , the line space is defined as the linear envelope of the line vectors . The dimension of the line space is called the line rank , it corresponds to the maximum number of linearly independent line vectors. Similarly, the column space and the column rank are defined by the column vectors. For matrices with entries from one field , one can show that the row and column rank of every matrix are the same, and therefore we speak of the (well-defined) rank of the matrix. This generally does not apply to matrices over rings . ${\displaystyle A}$ ${\displaystyle ZR(A)}$ ${\displaystyle A}$ ${\displaystyle SR(A)}$
• The rank of a system of finitely many vectors corresponds to the dimension of its linear envelope.
• In the case of a linear mapping , the rank is defined as the dimension of the image of this mapping: ${\displaystyle f}$ ${\displaystyle \mathrm {Bild} (f)}$

${\displaystyle \mathrm {rang} (f)=\dim(\mathrm {Bild} (f)).}$

A linear mapping and the associated mapping matrix have the same rank.

calculation

In order to determine the rank of a matrix, it is transformed into an equivalent matrix in (row) step form using the Gaussian elimination process. The number of row vectors that are not equal then corresponds to the rank of the matrix. ${\displaystyle 0}$

Examples:

• ${\displaystyle A={\begin{pmatrix}1&2&3\\0&5&4\\0&10&2\end{pmatrix}}\sim {\begin{pmatrix}1&2&3\\0&5&4\\0&0&-6\end{pmatrix}}\Rightarrow \mathrm {rang} (A)=3}$

• ${\displaystyle B={\begin{pmatrix}1&2&3\\0&6&4\\0&3&2\end{pmatrix}}\sim {\begin{pmatrix}1&2&3\\0&6&4\\0&0&0\end{pmatrix}}\Rightarrow \mathrm {rang} (B)=2}$

• ${\displaystyle C={\begin{pmatrix}2&3\\0&1\\4&-1\end{pmatrix}}\sim {\begin{pmatrix}2&3\\0&1\\0&0\end{pmatrix}}\Rightarrow \mathrm {rang} (C)=2}$

Alternatively, the matrix can also be reshaped in the form of columns. The rank of the matrix then corresponds to the number of column vectors that are not equal to 0.

Square matrices

If the rank of a square matrix is ​​the same as its number of rows and columns, it has full rank and is regular (invertible). This property can also be determined based on its determinant . A square matrix has full rank if and only if its determinant is different from zero or none of its eigenvalues ​​is zero.

properties

Be the following . ${\displaystyle m,n,l\in \mathbb {N} }$

• The only matrix with rank is the zero matrix  . The - identity matrix has full rank . ${\displaystyle 0}$ ${\displaystyle 0_{m,n}}$ ${\displaystyle n\!\times \!n}$ ${\displaystyle E_{n}}$ ${\displaystyle n}$
• The following applies to the rank of a matrix : ${\displaystyle m\!\times \!n}$ ${\displaystyle A}$

${\displaystyle \mathrm {rang} (A)\leq \min\{m,n\}.}$

The matrix is ​​said to have full rank if this inequality is equal.

• The transpose of a matrix has the same rank as : ${\displaystyle A^{T}}$ ${\displaystyle A}$ ${\displaystyle A}$

${\displaystyle \operatorname {rang} (A)=\operatorname {rang} (A^{T})\;.}$

• Extension: The rank of a matrix and the associated Gram matrix are the same if a real matrix is: ${\displaystyle A}$ ${\displaystyle A}$

  ${\displaystyle \mathrm {rang} (A)=\mathrm {rang} (A^{T}A)=\mathrm {rang} (AA^{T})=\mathrm {rang} (A^{T})\;.}$

• Subadditivity : For two matrices and the following applies: ${\displaystyle m\!\times \!n}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle \mathrm {rang} (A+B)\leq \mathrm {rang} (A)+\mathrm {rang} (B).}$

• Rank inequalities of Sylvester : For a matrix and a matrix we have: ${\displaystyle m\!\times \!n}$ ${\displaystyle A}$ ${\displaystyle n\!\times \!l}$ ${\displaystyle B}$

${\displaystyle \mathrm {rang} (A)+\mathrm {rang} (B)-n\leq \mathrm {rang} (A\cdot B)\leq \mathrm {min} \left\{\mathrm {rang} (A),\mathrm {rang} (B)\right\}.}$

• Condition according to Fontené , Rouché and Frobenius : A linear system of equations is solvable if and only if applies or (equivalent) . ${\displaystyle A\cdot x=b}$ ${\displaystyle b\in SR(A)}$ ${\displaystyle \mathrm {rang} (A)=\mathrm {rang} (A|b)}$
• A linear mapping is injective if and only if the mapping matrix has full column rank: ${\displaystyle A\in K^{m\times n}}$ ${\displaystyle \mathrm {rang} (A)=n.}$
• A linear mapping is surjective if and only if the mapping matrix has full row rank: ${\displaystyle A\in K^{m\times n}}$ ${\displaystyle \mathrm {rang} (A)=m.}$
• A linear mapping is bijective if and only if the mapping matrix is regular (invertible), because then the inverse mapping with mapping matrix exists . This is the case if and only if is quadratic ( ) and has full rank: ${\displaystyle A\in K^{m\times n}}$ ${\displaystyle A^{-1}}$ ${\displaystyle A}$ ${\displaystyle m=n}$ ${\displaystyle \mathrm {rang} (A)=m=n.}$
• Theorem for linear mappings : For the rank and defect (dimension of the core ) of a linear map from an n-dimensional vector space V to an m-dimensional vector space W, the relationship applies

${\displaystyle \dim V=\mathrm {rang} (f)+\mathrm {def} (f)\;.}$

literature

• Gerd Fischer: Linear Algebra. 13th edition. Vieweg, Braunschweig, Wiesbaden 2002, ISBN 3-528-97217-3 .

Individual evidence

1. ^ Serge Lang: Algebra 3rd edition. Springer, New York 2002, ISBN 0-387-95385-X .
2. ↑ Falko Lorenz: Lineare Algebra I. 3rd edition. Spectrum, Heidelberg 1992, ISBN 3-411-15193-5 .