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Encyclopedia > Minor (linear algebra)

In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. Suppose A is an mn matrix and k is a positive integer not larger than m and n. A kk minor of A is the determinant of a kk 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 kk.


Especially important are the (n-1)(n-1) minors of an nn 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 mn matrix with real entries (or entries from any other field) and rank r, then there exists at least one non-zero rr minor, while all larger minors are zero.


We will use the following notation for minors: if A is an mn 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 kk 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 mn matrix, B is an np 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 kk minors appear as the components of the resulting k-vectors. For example, the 22 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.


In graph theory, the term minor has a different, unrelated meaning. See minor (graph theory).


  Results from FactBites:
 
Minor (linear algebra) - Wikipedia, the free encyclopedia (552 words)
In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix.
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.
for the k×k minor of A that corresponds to the rows with index in I and the columns with index in J.
Minor (graph theory) - Wikipedia, the free encyclopedia (245 words)
In graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that results from G by zero or more edge contractions or edge deletions, or deletions of disconnected vertices.
The relation "being a minor of" is a partial order on the isomorphism classes of graphs.
In linear algebra, there is a different unrelated meaning of the word minor.
  More results at FactBites »

 
 

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