Minor (linear algebra)

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In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A.

Contents 1 Detailed definition 2 Nomenclature 3 Cofactors 4 Example 5 Applications 6 Multilinear algebra approach 7 References

[edit] 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

(read "m choose k")

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

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

minors of size k × k.

[edit] Nomenclature

A minor that is formed by removing only one row and column from a matrix A is called a first minor. When two rows and columns are now removed, this is called a second minor. ^[1]

[edit] 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.

[edit] Example

For example, given the matrix

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):

yields

where the vertical bars around the matrix indicate that the determinant should be taken. Thus, C₂₃ is (-1)²⁺³ M₂₃

[edit] 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

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.

[edit] 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

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

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

and

we can simplify this expression to

where the coefficients agree with the minors computed earlier.

[edit] References

v • d • e Topics related to linear algebra Vectors · Vector spaces · Linear span · Linear transformation · Linear independence · Linear combination · Basis · Column space · Row space · Dual space · Orthogonality · Rank · Minor · Eigenvalue, eigenvector and eigenspace  · Least squares regressions · Outer product · Cross product · Dot product · Transpose · Matrix decomposition