minor (of a matrix)

Given an n×m matrix A with entries aij, a minor of A is the determinantMathworldPlanetmath of a smaller matrix formed from its entries by selecting only some of the rows and columns. Let K={k1,k2,,kp} and L={l1,l2,,lp} be subsets of {1,2,,n} and {1,2,,m}, respectively. The indices are chosen such that k1<k2<<kp and l1<l2<<lp. The p-th order minor defined by K and L is the following determinant


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 conjunctionMathworldPlanetmath with the concept of a cofactorPlanetmathPlanetmath (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