Minor (linear algebra)

In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. Suppose A is an m×n matrix and k is a positive integer not larger than m and 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.

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

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.

For example, given the matrix

and suppose we wish to find the cofactor C₂₃. We consider the matrix with row 2 and column 3 removed (note the following is not standard notation!):

This gives:

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.

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

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

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 kxk minors appear as the components of the resulting k-vectors.

In graph theory, the term minor has a different, unrelated meaning. See minor (graph theory).