Sub-matrix

A sub-matrix is ​​created by deleting certain rows and columns of a matrix, here the second row and the fourth column.

In mathematics, a sub-matrix , also known as a sub-matrix or a deletion matrix , is a matrix that is created by deleting rows and columns from a given matrix. A sub-matrix of a square matrix in which the same rows and columns are deleted is also known as the main sub- matrix . Among other things, sub-matrices are used to define the minors and cofactors of a matrix. They play an important role in Laplace's expansion theorem of the determinant of a matrix.

definition

Is a matrix with the body , then is a sub-matrix of a matrix arises from the fact that the lines of the index set and the columns of the index set out are deleted, that is: ${\displaystyle A=(a_{ij})\in K^{m\times n}}$ ${\displaystyle K}$ ${\displaystyle A_{IJ}}$ ${\displaystyle A}$ ${\displaystyle I\subseteq \{1,\ldots ,m\}}$ ${\displaystyle J\subseteq \{1,\ldots ,n\}}$ ${\displaystyle A}$

${\displaystyle A_{IJ}=(a_{ij})_{i\in \{1,\ldots ,m\}\setminus I,j\in \{1,\ldots ,n\}\setminus J}}$

The sub-matrix then has rows and columns. In the case of single-element index sets, write briefly instead of . If and are, becomes a sub-matrix ${\displaystyle A_{IJ}}$ ${\displaystyle m-|I|}$ ${\displaystyle n-|J|}$ ${\displaystyle A_{ij}}$ ${\displaystyle A_{\{i\}\{j\}}}$ ${\displaystyle m=n}$ ${\displaystyle I=J}$

${\displaystyle A_{I}=A_{II}}$   or.   ${\displaystyle A_{i}=A_{ii}}$

also known as the main sub-matrix. Occasionally a sub-matrix is ​​also noted by specifying the rows and columns that make up it as indices. One then writes:

${\displaystyle A_{IJ}=(a_{ij})_{i\in I,j\in J}}$

In the following, however, the former notation variant is used. Sub-matrices, which are made up of consecutive row and column indices, form a block of a matrix.

example

The real matrix is ​​given

${\displaystyle A={\begin{pmatrix}1&2&3&4\\5&6&7&8\\9&10&11&12\end{pmatrix}}\in \mathbb {R} ^{3\times 4}}$,

then is the sub-matrix

${\displaystyle A_{23}=A_{\{2\}\{3\}}={\begin{pmatrix}1&2&4\\9&10&12\end{pmatrix}}\in \mathbb {R} ^{2\times 3}}$

the matrix that results from deleting the second row and the third column.

use

Each matrix with rank has a square sub-matrix so that ${\displaystyle A\in K^{m\times n}}$ ${\displaystyle r}$ ${\displaystyle A_{IJ}\in K^{r\times r}}$

${\displaystyle \operatorname {rang} (A_{IJ})=\operatorname {rang} (A)}$

holds and its determinant

${\displaystyle \operatorname {det} (A_{IJ})\neq 0}$

is. Such a sub-matrix can be found, for example, with the aid of the Gaussian elimination method. The determinant of a square sub-matrix is ​​also known as the minor or sub-determinant. The determinant of a main sub-matrix is ​​accordingly called the main minor. The determinants of the sub-matrices of a square matrix are given alternating signs cofactors ${\displaystyle A_{ij}}$ ${\displaystyle A}$

${\displaystyle {\tilde {a}}_{ij}=(-1)^{i+j}\operatorname {det} (A_{ij})}$

called the matrix. With the help of the cofactor matrix , the inverse of the matrix can be specified explicitly. Sub-matrices also play an important role in Laplace's expansion theorem of the determinant of a matrix and in Binet-Cauchy's theorem for determining the determinant of the product of two matrices. ${\displaystyle {\tilde {A}}=({\tilde {a}}_{ij})}$ ${\displaystyle A}$

literature

• Siegfried Bosch : Linear Algebra . Springer, 2006, ISBN 3-540-29884-3 .
• Christoph W. Überhuber: Computer Numerics 2 . Springer, 1995, ISBN 3-642-57794-6 .

Individual evidence

1. ^ Christian Karpfinger: Higher mathematics in recipes. Springer Verlag, Berlin 2014, ISBN 978-3-642-37865-2 , p. 95.
2. ↑ Christoph Überhuber: Computer Numerics 2 . S. 212 .
3. ^ Bosch: Linear Algebra . S. 146 .

Web links

• TS Pigolkina: Submatrix . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Submatrix . In: MathWorld (English).
• Mathprof: Submatrix notation . In: PlanetMath . (English)