minor (of a matrix)

Given an $n\times m$ matrix $A$ with entries $a_{ij}$ , a minor of $A$ is the determinant of a smaller matrix formed from its entries by selecting only some of the rows and columns. Let $K=\{k_{1},k_{2},\ldots,k_{p}\}$ and $L=\{l_{1},l_{2},\ldots,l_{p}\}$ be subsets of $\{1,2,\ldots,n\}$ and $\{1,2,\ldots,m\}$ , respectively. The indices are chosen such that $k_{1}<k_{2}<\cdots<k_{p}$ and $l_{1}<l_{2}<\cdots<l_{p}$ . The $p$ -th order minor defined by $K$ and $L$ is the following determinant

A\begin{pmatrix}k_{1}&k_{2}&\cdots&k_{p}\\ l_{1}&l_{2}&\cdots&l_{p}\end{pmatrix}=\begin{vmatrix}a_{k_{1}l_{1}}&a_{k_{1}l_% {2}}&\cdots&a_{k_{1}l_{p}}\\ a_{k_{2}l_{1}}&a_{k_{2}l_{2}}&\cdots&a_{k_{2}l_{p}}\\ \vdots&\vdots&\ddots&\vdots\\ a_{k_{p}l_{1}}&a_{k_{p}l_{2}}&\cdots&a_{k_{p}k_{p}}\end{vmatrix}.

If $p$ exceeds either $m$ or $n$ , then the minor is automatically zero. When $p=m=n$ , the minor is simply the determinant of the matrix. If $K=L$ , then the minor is called principal. The word minor may also refer to just the matrix formed from the selected rows and columns, not necessarily its determinant. The precise meaning is usually clear from context.

There does not seem to be a standard notation for matrix minors. Another possible notation is $[A]_{K,L}$ .

Some authors reserve the term minor for the case when only one row and one column are removed. This use is in conjunction with the concept of a cofactor (http://planetmath.org/LaplaceExpansion).

Title	minor (of a matrix)
Canonical name	MinorofAMatrix
Date of creation	2013-03-22 14:07:01
Last modified on	2013-03-22 14:07:01
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 15A15
Related topic	LaplaceExpansion
Related topic	CauchyBinetFormula
Defines	principal minor
Defines	cofactor