# Minor (linear algebra)

﻿
Minor (linear algebra)

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which in turn are useful for computing both the determinant and inverse of square matrices.

## Detailed definition

Let A be an m × n matrix and k an integer with 0 < km, and kn. A k × k minor of A is the determinant of a k × k matrix obtained from A by deleting mk rows and nk columns.

Since there are: ${m \choose k}$ (read "m choose k")

ways to choose k rows from m rows, and there are ${n \choose k}$

ways to choose k columns from n columns, there are a total of ${m \choose k} \cdot {n \choose k}$

minors of size k × k.

## Nomenclature

The (i,j) minor (often denoted Mij) of an n × n square matrix A is defined as the determinant of the (n − 1) × (n − 1) matrix formed by removing from A its ith row and jth column. An (i,j) minor is also referred to as (i,j)th minor, or simply i,j minor.

Mij is also called the minor of the element aij of matrix A.

A minor that is formed by removing only one row and column from a square matrix A (such as Mij) is called a first minor. When two rows and columns are removed, this is called a second minor.

The (i,j) cofactor Cij of a square matrix A is just (−1)i + j times the corresponding (n − 1) × (n − 1) minor Mij:

Cij = (−1)i + j Mij

The cofactor matrix of A, or matrix of A cofactors, typically denoted C, is defined as the n×n matrix whose (i,j) entry is the (i,j) cofactor of A.

The transpose of C is called the adjugate or classical adjoint of A. (In modern terminology, the "adjoint" of a matrix most often refers to the corresponding adjoint operator.) Adjugate matrices are used to compute the inverse of square matrices.

## Example

For example, given the matrix $\begin{pmatrix} \,\,\,1 & 4 & 7 \\ \,\,\,3 & 0 & 5 \\ -1 & 9 & \!11 \\ \end{pmatrix}$

suppose we wish to find the cofactor C23. The minor M23 is the determinant of the above matrix with row 2 and column 3 removed (the following is not standard notation): $\begin{vmatrix} \,\,1 & 4 & \Box\, \\ \,\Box & \Box & \Box\, \\ -1 & 9 & \Box\, \\ \end{vmatrix}$ yields $\begin{vmatrix} \,\,\,1 & 4\, \\ -1 & 9\, \\ \end{vmatrix} = (9-(-4)) = 13$

where the vertical bars around the matrix indicate that the determinant should be taken. Thus, C23 is (-1)2+3 M23 $= -13. \!\$

## Complement

The complement, C, of a minor, M, of a square matrix, A, is formed by the determinant of the matrix A from which all the rows and columns associated with M have been removed. The complement of the first minor of an element aij is merely that element.

## Applications

The cofactors feature prominently in Laplace's formula for the expansion of determinants. If all the cofactors of a square matrix A are collected to form a new matrix of the same size and then transposed, one obtains the adjugate of A, which is useful in calculating the inverse of small matrices.

Given an m × n matrix with real entries (or entries from any other field) and rank r, then there exists at least one non-zero r × r minor, while all larger minors are zero.

We will use the following notation for minors: if A is an m × n matrix, I is a subset of {1,...,m} with k elements and J is a subset of {1,...,n} with k elements, then we write [A]I,J for the k × k minor of A that corresponds to the rows with index in I and the columns with index in J.

• If I = J, then [A]I,J is called a principal minor.
• If the matrix that corresponds to a principal minor is a quadratic upper-left part of the larger matrix (i.e., it consists of matrix elements in rows and columns from 1 to k), then the principal minor is called a leading principal minor. For an n × n square matrix, there are n leading principal minors. (Not in agreement with a lot of books. Sometimes the leading principal minor is consider to be the leading k x k matrix.)
• For Hermitian matrices, the principal minors can be used to test for positive definiteness.

Both the formula for ordinary matrix multiplication and the Cauchy-Binet formula for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices. Suppose that A is an m × n matrix, B is an n × p matrix, I is a subset of {1,...,m} with k elements and J is a subset of {1,...,p} with k elements. Then $[\mathbf{AB}]_{I,J} = \sum_{K} [\mathbf{A}]_{I,K} [\mathbf{B}]_{K,J}\,$

where the sum extends over all subsets K of {1,...,n} with k elements. This formula is a straightforward extension of the Cauchy-Binet formula.

## Multilinear algebra approach

A more systematic, algebraic treatment of the minor concept is given in multilinear algebra, using the wedge product: the k-minors of a matrix are the entries in the kth exterior power map.

If the columns of a matrix are wedged together k at a time, the k × k minors appear as the components of the resulting k-vectors. For example, the 2 × 2 minors of the matrix $\begin{pmatrix} 1 & 4 \\ 3 & \!\!-1 \\ 2 & 1 \\ \end{pmatrix}$

are −13 (from the first two rows), −7 (from the first and last row), and 5 (from the last two rows). Now consider the wedge product $(\mathbf{e}_1 + 3\mathbf{e}_2 +2\mathbf{e}_3)\wedge(4\mathbf{e}_1-\mathbf{e}_2+\mathbf{e}_3)$

where the two expressions correspond to the two columns of our matrix. Using the properties of the wedge product, namely that it is bilinear and $\mathbf{e}_i\wedge \mathbf{e}_i = 0$

and $\mathbf{e}_i\wedge \mathbf{e}_j = - \mathbf{e}_j\wedge \mathbf{e}_i,$

we can simplify this expression to $-13 \mathbf{e}_1\wedge \mathbf{e}_2 -7 \mathbf{e}_1\wedge \mathbf{e}_3 +5 \mathbf{e}_2\wedge \mathbf{e}_3$

where the coefficients agree with the minors computed earlier.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Linear algebra — R3 is a vector (linear) space, and lines and planes passing through the origin are vector subspaces in R3. Subspaces are a common object of study in linear algebra. Linear algebra is a branch of mathematics that studies vector spaces, also called …   Wikipedia

• Cofactor (linear algebra) — In linear algebra, the cofactor (sometimes called adjunct, see below) describes a particular construction that is useful for calculating both the determinant and inverse of square matrices. Specifically the cofactor of the (i, j) entry of a… …   Wikipedia

• Rank (linear algebra) — The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A. Equivalently, the column rank of A is the dimension of the …   Wikipedia

• Projection (linear algebra) — Orthogonal projection redirects here. For the technical drawing concept, see orthographic projection. For a concrete discussion of orthogonal projections in finite dimensional linear spaces, see vector projection. The transformation P is the… …   Wikipedia

• Basis (linear algebra) — Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference. In linear algebra, a basis is a set of linearly independent vectors that, in a linear… …   Wikipedia

• List of linear algebra topics — This is a list of linear algebra topics. See also list of matrices glossary of tensor theory. Contents 1 Linear equations 2 Matrices 3 Matrix decompositions 4 …   Wikipedia

• Minor — Not to be confused with myna or miner. Contents 1 Mathematics 2 Music 3 Surname …   Wikipedia

• Linear map — In mathematics, a linear map, linear mapping, linear transformation, or linear operator (in some contexts also called linear function) is a function between two vector spaces that preserves the operations of vector addition and scalar… …   Wikipedia

• Linear Complementarity Problem — Das lineare Komplementaritätsproblem (LKP, engl. linear complementarity problem) ist ein mathematisches Problem aus der Linearen Algebra. Gegeben sei eine rationale Matrix und ein rationaler Vektor , dann finde Vektoren so, dass die drei… …   Deutsch Wikipedia

• Closed linear span — In functional analysis, a branch of mathematics, the closed linear span of a set of vectors is the minimal closed set which contains the linear span of that set. Contents 1 Definition 2 Notes 3 A useful lemma …   Wikipedia

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.