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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 ''i''th row and the ''j''th 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 : suppose we wish to find the cofactor ''C''23. The minor ''M''23 is the determinant of the above matrix with row 2 and column 3 removed (the following is not standard notation): : yields Thus ''C''23 is ''M''23 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 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 : 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 : 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. |
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