Minor (linear algebra)

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.

Detailed definition

Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ 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

_mC_k    (read "m choose k")

ways from m rows to choose k rows, and there are

_nC_k

ways from n columns to choose k columns, there are a total of

_mC_k · _nC_k

minors of size k × k.

Cofactors

The (n − 1) × (n − 1) minor (often denoted M_ij) of an n × n square matrix is defined as the determinant of the matrix formed by removing the i^th row and the j^th column.

The cofactor C_ij of a square matrix A is just (−1)^{i + j} times the corresponding minor:

C_ij = (−1)^{i + j} M_ij

The transpose of the matrix C of cofactors is called the adjugate matrix.

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\!\$

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.
• 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 corollary 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.