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 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

<math>\begin{pmatrix}

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

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):

<math> \begin{vmatrix}

1 & 4 & \Box \\ \Box & \Box & \Box \\ -1 & 9 & \Box \\ \end{vmatrix}</math> yields <math>\begin{vmatrix} 1 & 4 \\ -1 & 9 \\ \end{vmatrix} = (9-(-4)) = 13.</math>

Thus C₂₃ is <math> (-1)^{2+3} \!\ </math> M₂₃ <math> = -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. If I=J, then [A]_I,J is called a principal minor.

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 straightforward 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

<math>\begin{pmatrix}

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>

and

<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.ar:مختصر (جبر خطي) de:Minor (Mathematik) nl:Minor pl:Minor macierzy