Minor (linear algebra)

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This article is about a concept in linear algebra. For the unrelated concept of "minor" in graph theory, see minor (graph theory).

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A.

1 Detailed definition
2 Cofactors
3 Example
4 Applications
5 Multilinear algebra approach

[edit] Detailed definition

Let A be an m × n matrix and k is a positive integer not greater then min(m, n). A k × k minor of A is the determinant of a k × k matrix obtained from A by deleting m - k rows and n - k columns.

Since there are C(m, k) choices of k rows out of m, and there are C(n, k) choices of k columns out of n, there are a total of C(m, k)C(n, k) minors of size k × k.

[edit] Cofactors

Especially important are the minors, that are (n - 1) × (n - 1), of an n × n square matrix - these are often denoted M_ij, and are derived by removing the ith row and the jth column and then computing the determinant.

The cofactors of a square matrix A are closely related to the minors of A: the cofactor C_ij of A is defined as (−1)^{i + j} times the minor M_ij of A.

[edit] 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 \!\$

[edit] 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, 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 includes the upper-left corner of the larger matrix (i.e., the matrix element in first row and first column), then the principal minor is called a leading principal minor. For an n×n square matrix, there are n leading principal minors.
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

$[AB]_{I,J} = \sum_{K} [A]_{I,K} [B]_{K,J}\,$

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

[edit] Multilinear algebra approach

A more systematic, algebraic treatment of the minor concept is given in multilinear algebra, using the wedge product. 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

$\begin{pmatrix} 1 & 4 \\ 3 & -1 \\ 2 & 1 \\ \end{pmatrix}$