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 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


1 & 4 & 7 \\ 3 & 0 & 5 \\ -1 & 9 & 11 \\ \end{pmatrix}<math>

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!):


1 & 4 & ! \\ ! & ! & ! \\ -1 & 9 & ! \\ \end{pmatrix}<math>

This gives:


1 & 4 \\ -1 & 9 \\ \end{vmatrix}=(-1)(9+4)=-13.<math>

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

<math>[AB]_{I,J} = \sum_{K} [A]_{I,K} [B]_{K,J}\,<math>

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 k×k minors appear as the components of the resulting k-vectors. For example, the 2×2 minors of the matrix


1 & 4 \\ 3 & -1 \\ 2 & 1 \\ \end{pmatrix}<math> 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

<math>(\mathbf{e}_1 + 3\mathbf{e}_2 +2\mathbf{e}_3)\wedge(4\mathbf{e}_1-\mathbf{e}_2+\mathbf{e}_3)<math>

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

<math>\mathbf{e}_i\wedge \mathbf{e}_i = 0<math>


<math>\mathbf{e}_i\wedge \mathbf{e}_j = - \mathbf{e}_j\wedge \mathbf{e}_i,<math>

we can simplify this expression to

<math> -13 \mathbf{e}_1\wedge \mathbf{e}_2 -7 \mathbf{e}_1\wedge \mathbf{e}_3 +5 \mathbf{e}_2\wedge \mathbf{e}_3<math>

where the coefficients agree with the minors computed earlier.

In graph theory, the term minor has a different, unrelated meaning. See minor (graph theory).de:Minor (Mathematik) pl:Minor macierzy


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