Minor (linear algebra)
In linear algebra, a minor of a matrix is the determinant of some smaller square matrix generated from by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices are required for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition.
Definition and illustration
First minors
If is a square matrix, then the minor of the entry in the -th row and -th column is the determinant of the submatrix formed by deleting the -th row and -th column. This number is often denoted. The cofactor is obtained by multiplying the minor by.To illustrate these definitions, consider the following matrix,
To compute the minor and the cofactor, we find the determinant of the above matrix with row 2 and column 3 removed.
So the cofactor of the entry is
General definition
Let be an matrix and an integer with, and. A minor of, also called minor determinant of order of or, if, the th ''minor determinant of is the determinant of a matrix obtained from by deleting rows and columns. Sometimes the term is used to refer to the matrix obtained from as above, but this matrix should be referred to as a submatrix of, leaving the term "minor" to refer to the determinant of this matrix. For a matrix as above, there are a total of minors of size. The minor of order zero is often defined to be 1. For a square matrix, the zeroth minor'' is just the determinant of the matrix.Let
be ordered sequences of indexes. The minor corresponding to these choices of indexes is denoted or or or or or , depending on the source. Also, there are two types of denotations in use in literature: by the minor associated to ordered sequences of indexes and, some authors mean the determinant of the matrix that is formed as above, by taking the elements of the original matrix from the rows whose indexes are in and columns whose indexes are in, whereas some other authors mean by a minor associated to and the determinant of the matrix formed from the original matrix by deleting the rows in and columns in ; which notation is used should always be checked. In this article, we use the inclusive definition of choosing the elements from rows of and columns of. The exceptional case is the case of the first minor or the -minor described above; in that case, the exclusive meaning is standard everywhere in the literature and is used in this article also.
Complement
The complement of a minor of a square matrix,, is formed by the determinant of the matrix from which all the rows and columns associated with have been removed. The complement of the first minor of an element is merely that element.Applications of minors and cofactors
Cofactor expansion of the determinant
The cofactors feature prominently in Laplace's formula for the expansion of determinants, which is a method of computing larger determinants in terms of smaller ones. Given an matrix, the determinant of, denoted, can be written as the sum of the cofactors of any row or column of the matrix multiplied by the entries that generated them. In other words, defining then the cofactor expansion along the -th column gives:The cofactor expansion along the -th row gives:
Inverse of a matrix
One can write down the inverse of an invertible matrix by computing its cofactors by using Cramer's rule, as follows. The matrix formed by all of the cofactors of a square matrix is called the cofactor matrix :Then the inverse of is the transpose of the cofactor matrix times the reciprocal of the determinant of :
The transpose of the cofactor matrix is called the adjugate matrix of.
The above formula can be generalized as follows: Let
be ordered sequences of indexes. Then
where denote the ordered sequences of indices complementary to, so that every index appears exactly once in either or, but not in both and denotes the determinant of the submatrix of formed by choosing the rows of the index set and columns of index set. Also, A simple proof can be given using wedge product. Indeed,
where are the basis vectors. Acting by on both sides, one gets
The sign can be worked out to be
so the sign is determined by the sums of elements in and.
Other applications
Given an matrix with real entries and rank, then there exists at least one non-zero minor, while all larger minors are zero.We will use the following notation for minors: if is an matrix, is a subset of with elements, and is a subset of with elements, then we write for the minor of that corresponds to the rows with index in and the columns with index in.
- If, then is called a principal minor.
- If the matrix that corresponds to a principal minor is a square upper-left submatrix of the larger matrix, then the principal minor is called a leading principal minor or corner minor . For an square matrix, there are leading principal minors.
- A basic minor of a matrix is the determinant of a square submatrix that is of maximal size with nonzero determinant.
- For Hermitian matrices, the leading principal minors can be used to test for positive definiteness and the principal minors can be used to test for positive semidefiniteness. See Sylvester's criterion for more details.
Suppose that is an matrix, is an matrix, is a subset of with elements and is a subset of with elements. Then
where the sum extends over all subsets of with elements. This formula is a straightforward extension of the Cauchy–Binet formula.
Multilinear algebra approach
A more systematic, algebraic treatment of minors is given in multilinear algebra, using the wedge product: the -minors of a matrix are the entries in the -th exterior power map.If the columns of a matrix are wedged together at a time, the minors appear as the components of the resulting -vectors. For example, the 2 × 2 minors of the matrix
are −13, −7, and 5. 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 alternating,
and antisymmetric,
we can simplify this expression to
where the coefficients agree with the minors computed earlier.
A remark about different notation
In some books, instead of cofactor the term adjunct is used. Moreover, it is denoted as and defined in the same way as cofactor:Using this notation the inverse matrix is written this way:
Keep in mind that adjunct is not adjugate or adjoint. In modern terminology, the "adjoint" of a matrix most often refers to the corresponding adjoint operator.