Defect (math)

In mathematics, the defect is a term from the subfield of linear algebra . You assign it to a linear mapping or a matrix .

Definition of linear mapping

Let and be two finite-dimensional vector spaces , the dimension of be , the dimension of be . Further be a linear map. Then the defect of this map is defined as the dimension of the core of the map, in short ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle W}$ ${\displaystyle m}$ ${\displaystyle f\colon V\to W}$

${\displaystyle \operatorname {def} (f)=\operatorname {dim} (\operatorname {ker} (f))}$.

Defect in matrices

A matrix with elements from one body can be interpreted as a linear mapping . In this sense, the defect is caused by the matrix ${\displaystyle A\in \mathbb {K} ^{m\times n}}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle f_{A}:\mathbb {K} ^{n}\to \mathbb {K} ^{m},x\mapsto Ax}$ ${\displaystyle A}$

${\displaystyle \operatorname {def} (A):=\operatorname {def} (f_{A})}$

Are defined. The defect of is therefore equal to the dimension of the solution space of the homogeneous linear system of equations . ${\displaystyle A}$ ${\displaystyle Ax=0}$

If  the matrix is ​​zero , then is  equal to the number of columns of . Otherwise is equal to the maximum number of columns that can be deleted so that the reduced matrix has the same picture as . The deleted columns are then linearly dependent on the columns remaining in the reduced matrix. ${\displaystyle A}$ ${\displaystyle \operatorname {def} (A)}$ ${\displaystyle A}$ ${\displaystyle \operatorname {def} (A)}$ ${\displaystyle A}$ ${\displaystyle A}$

calculation

The Gaussian elimination method with exchanging rows and columns is particularly suitable for manual calculations with small matrices to determine the defect. With this method, each matrix can be transformed into an equivalent matrix with for , in which the diagonal elements of the first rows are occupied by non-zero elements and the remaining rows are zero rows ( is the rank of the matrix ). The defect of this matrix is ​​then (that is the statement of the ranking ). ${\displaystyle A\in \mathbb {K} ^{m\times n}}$ ${\displaystyle {\bar {A}}}$ ${\displaystyle {\bar {A}}_{i,j}=0}$ ${\displaystyle i>j}$ ${\displaystyle r\in \{0,\ldots ,m\}}$ ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle A}$ ${\displaystyle \operatorname {def} (A)=n-r}$

Assume that is not the zero matrix. It sweeps from those columns, the columns in the matrix correspond (this is during the Gaussian elimination column transpositions carried out to take into account), the reduced matrix has the same image as . If further columns are deleted (if this is possible) the image of the matrix is ​​reduced. ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle r+1,\ldots ,n}$ ${\displaystyle {\bar {A}}}$ ${\displaystyle A}$

In the case of square matrices (i.e. for ) the defect of is equal to the number of zero lines in . ${\displaystyle m=n}$ ${\displaystyle A}$ ${\displaystyle {\bar {A}}}$

Determining the defect of a matrix by means of singular value decomposition is numerically more stable, but also more complex than the Gaussian elimination method .

Examples

${\displaystyle A={\begin{pmatrix}1&2&3\\0&5&4\\0&10&2\end{pmatrix}}\sim {\bar {A}}={\begin{pmatrix}1&2&3\\0&5&4\\0&0&-6\end{pmatrix}}\Rightarrow n=3,r=3\Rightarrow \mathrm {def} (A)=0}$

${\displaystyle A={\begin{pmatrix}1&2&3\\0&6&4\\0&3&2\end{pmatrix}}\sim {\bar {A}}={\begin{pmatrix}1&2&3\\0&6&4\\0&0&0\end{pmatrix}}\Rightarrow n=3,r=2\Rightarrow \mathrm {def} (A)=1,}$

It wasn't necessary to swap columns, so the matrix has

${\displaystyle {\begin{pmatrix}1&2\\0&6\\0&3\end{pmatrix}},}$

which results from deleting the last column, the same picture as . ${\displaystyle A}$ ${\displaystyle A}$

${\displaystyle A={\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}}\sim {\bar {A}}={\begin{pmatrix}1&2&3\\0&-3&-6\end{pmatrix}}\Rightarrow n=3,r=2\Rightarrow \operatorname {def} (A)=1}$

Again, it was not necessary to swap columns, so this matrix has the same picture as ${\displaystyle {\begin{pmatrix}1&2\\4&5\end{pmatrix}}.}$

${\displaystyle A={\begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}}\sim {\bar {A}}={\begin{pmatrix}1&2\\0&-2\\0&0\end{pmatrix}}\Rightarrow n=2,r=2\Rightarrow \operatorname {def} (A)=0}$

Ranking

Main article: Rank set

The ranking shows a relationship between the defect and the rank of a linear mapping . ${\displaystyle \operatorname {rg} (f)}$ ${\displaystyle f\colon V\to W}$

${\displaystyle \dim V=\operatorname {def} (f)+\operatorname {rg} (f)}$

Individual evidence

1. ↑ Michael Artin: Algebra. Birkhäuser, Basel et al. 1998, ISBN 3-7643-5938-2 , p. 123 ( limited preview in the Google book search).