minor (of a matrix)

Given an $n\\times m$ matrix $A$ with entries $a_{ij}$ , a minor of $A$ isthe determinant of a smaller matrix formed from its entries by selectingonly 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 $p$ -thorder 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 isautomatically zero. When $p=m=n$ , the minor is simply the determinantof the matrix. If $K=L$ , then the minor is called principal.The word minor may also refer to just the matrix formed fromthe selected rows and columns, not necessarily its determinant. The precisemeaning 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 onerow and one column are removed. This use is in conjunction with theconcept of a cofactor (http://planetmath.org/LaplaceExpansion).