Minor (linear algebra)

Minor or sub-determinant is a term from the mathematical branch of linear algebra . It denotes the determinant of a square sub-matrix , which is created by deleting one or more columns and rows of a matrix . The number of rows or columns of the corresponding sub-matrix indicates the order of the minor.

Cofactors

definition

The cofactors (or cofactors ) for a square matrix are defined by the following formula: ${\displaystyle n\times n}$ ${\displaystyle A=(a_{ij})_{ij}}$ ${\displaystyle {\tilde {a}}_{ij}}$

${\displaystyle {\tilde {a}}_{ij}=(-1)^{i+j}\cdot M_{ij}}$

It is the minor -th order of those as a determinant sub-matrix is calculated that by deleting the th row and formed -th column. ${\displaystyle M_{ij}}$ ${\displaystyle (n-1)}$ ${\displaystyle i}$ ${\displaystyle j}$

Instead of deleting rows and columns, one can also consider matrices in which the entries in the -th row or the -th column (or both) are replaced by zeros, with the exception of the entry at the position that is replaced by a 1. We then get for the cofactors: ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle (i,j)}$

${\displaystyle {\tilde {a}}_{ij}={\begin{vmatrix}a_{1,1}&\dots &a_{1,j-1}&0&a_{1,j+1}&\dots &a_{1,n}\\\vdots &\ddots &\vdots &\vdots &\vdots &&\vdots \\a_{i-1,1}&\dots &a_{i-1,j-1}&0&a_{i-1,j+1}&\dots &a_{i-1,n}\\0&\dots &0&1&0&\dots &0\\a_{i+1,1}&\dots &a_{i+1,j-1}&0&a_{i+1,j+1}&\dots &a_{i+1,n}\\\vdots &&\vdots &\vdots &\vdots &\ddots &\vdots \\a_{n,1}&\dots &a_{n,j-1}&0&a_{n,j+1}&\dots &a_{n,n}\end{vmatrix}}}$

where stands for the formation of the determinant. A matrix, the cofactor matrix or komatrix , can be formed from the cofactors , the transpose of which is called adjuncts or complementary matrix. With it you can calculate the inverse of a matrix. The Laplace development kit uses a matrix to calculate their cofactors determinant. ${\displaystyle |\cdot |}$ ${\displaystyle n\times n}$

example

The minor and the cofactor of the following matrix are to be determined: ${\displaystyle M_{2,3}}$ ${\displaystyle {\tilde {a}}_{2,3}}$

${\displaystyle A={\begin{pmatrix}1&4&7\\3&0&5\\-1&9&11\end{pmatrix}}}$

By deleting the second row and third column

${\displaystyle {\begin{pmatrix}1&4&\Box \\\Box &\Box &\Box \\-1&9&\Box \end{pmatrix}}}$

the matrix is ​​created

${\displaystyle A_{2,3}={\begin{pmatrix}1&4\\-1&9\end{pmatrix}}}$

From this the minor can be calculated. ${\displaystyle M_{2,3}}$

${\displaystyle M_{2,3}={\begin{vmatrix}1&4\\-1&9\end{vmatrix}}=9+4=13.}$

The following applies to the cofactor ${\displaystyle {\tilde {a}}_{2,3}}$

${\displaystyle {\tilde {a}}_{2,3}=(-1)^{2+3}\cdot M_{2,3}=-13}$

or.

${\displaystyle {\tilde {a}}_{2,3}={\begin{vmatrix}1&4&0\\0&0&1\\-1&9&0\end{vmatrix}}=-13}$

Major Minors

definition

If minors are created by deleting rows and columns of the same number, one speaks of main minors , more precisely of main minors of the kth order , if the size of the sub-matrix is ​​to be specified. If exactly the first k rows and columns remain, one speaks of leading major minors of the kth order . The leading main minors are sometimes also called naturally ordered main minors . In German-speaking countries that are leading principal minors shortened often the principal minors called. This is particularly related to the fact that not all major minors have to be examined for many applications. In addition, the term main section determinant for the main minors is used in German-speaking countries .

To illustrate this, make it clear how many minors, main minors and leading main minors there are in a 3x3 matrix. If you first delete the i-th row and i-th column for i = 1,2,3, 3 main minors of the second order remain. If you delete several lines and the columns with the same number, in this case you do so with two, there are 3 main minors of the first order. The more lines are deleted, the smaller the order.

The main minors have a meaning due to the main minor criterion for determining the definiteness of symmetrical or Hermitian matrices .

Example of major minors and leading major minors

Leading main minors are special main minors that are created by successively shortening the starting matrix "from its end" by one row and column and calculating the determinants of the resulting sub-matrices. For example, the 3 × 3 matrix provides

${\displaystyle A={\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}}}$

the following 3 sub-matrices

${\displaystyle A_{1}={\begin{pmatrix}1\end{pmatrix}},\quad A_{2}={\begin{pmatrix}1&2\\4&5\\\end{pmatrix}},\quad A_{3}={\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}},}$

from which the following 3 leading major minors can then be calculated:

• Leading major minor 1st order: ${\displaystyle \det(A_{1})=1;}$
• Leading major minor, 2nd order: ${\displaystyle \det(A_{2})={\begin{vmatrix}1&2\\4&5\\\end{vmatrix}}=-3;}$
• Leading major minor 3rd order: ${\displaystyle \det(A_{3})={\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}}=0\,.}$

As you can see, there is only one major 3rd order minor, which is also the leading minor , namely the determinant of the entire matrix. In the case of the above starting matrix, further main minors that play a role in particular when determining the semi-definiteness of a matrix would be the following four main minors of the 1st and 2nd order:

• Further main minors 1st order: ${\displaystyle \det(a_{22})=5;\quad \det(a_{33})=9;}$
• Further main minors 2nd order: ${\displaystyle \det {\begin{pmatrix}a_{11}&a_{13}\\a_{31}&a_{33}\\\end{pmatrix}}={\begin{vmatrix}1&3\\7&9\\\end{vmatrix}}=-12;\quad \det {\begin{pmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\\\end{pmatrix}}={\begin{vmatrix}5&6\\8&9\\\end{vmatrix}}=-3\,.}$

Individual evidence

1. ^ Siegfried Bosch : Linear Algebra. Springer, 2001, ISBN 3-540-41853-9 , p. 148
2. ^ Frank Riedel: Mathematics for Economists . Jumper; Edition: 2nd verb. 2009 edition (September 28, 2009). ISBN 978-3642036484 . P. 220
3. ^ ^{A b} Alpha C. Chiang, Kevin Wainwright, Harald Nitsch: Mathematics for Economists - Basics, Methods and Applications . Vahlen; Edition: 1st edition. (January 2011). ISBN 978-3800636631 . Page 80
4. ↑ For example: Norbert Herrmann : Höhere Mathematik: for engineers, physicists and mathematicians . Oldenbourg Wissenschaftsverlag; Edition: 2nd revised. Edition (September 1, 2007). ISBN 978-3486584479 . Page 13
5. ^ Boker, Fred. Formula collection for economists: mathematics and statistics. Pearson Deutschland GmbH, 2007. p. 194.

literature

• Wolfgang Gawronski: Basics of Linear Algebra. Aula-Verlag, Wiesbaden 1996, ISBN 3-89104-566-2 , p. 193