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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 Mij, 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
Cij of A is defined as (-1)i+j times the minor
Mij 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).
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