Minor (linear algebra) 262107 222073275 2008-06-27T12:51:53Z Giftlite 37986 /* Example */ -. {{About|a concept in linear algebra|the unrelated concept of "minor" in graph theory|Minor (graph theory)}} In [[linear algebra]], a '''minor''' of a [[matrix (mathematics)|matrix]] '''A''' is the [[determinant]] of some smaller [[square matrix]], cut down from '''A''' by removing one or more of its rows or columns. Minors obtained by removing just one row and one column from square matrices ('''first minors''') are required for calculating matrix [[cofactor]]s, which in turn are useful for computing both the determinant and [[Inverse matrix|inverse]] of square matrices. ==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 :[[binomial coefficient|<math>{m \choose k}</math>]] (read "[[Binomial coefficient|''m'' choose ''k'']]") ways to choose ''k'' rows from ''m'' rows, and there are :<math>{n \choose k}</math> ways to choose ''k'' columns from ''n'' columns, there are a total of :<math>{m \choose k} \cdot {n \choose k}</math> minors of size ''k'' × ''k''. ==Nomenclature== The (''i'',''j'') minor (often denoted '''M'''<sub>''ij''</sub>) of an ''n'' × ''n'' [[square matrix]] '''A''' is defined as the determinant of the (''n'' − 1) × (''n'' − 1) matrix formed by removing from '''A''' its ''i''<sup>th</sup> row and ''j''<sup>th</sup> column. An (''i'',''j'') minor is also referred to as (''i'',''j'')<sup>th</sup> minor, or simply ''i'',''j'' minor. A minor that is formed by removing only one row and column from a square matrix '''A''' (such as '''M'''<sub>''ij''</sub>) is called a ''first minor''. When two rows and columns are removed, this is called a ''second minor''.<ref>Burnside, William Snow & Panton, Arthur William (1886) ''Theory of Equations: with an Introduction to the Theory of Binary Algebraic Form'', http://books.google.com/books?id=BhgPAAAAIAAJ&pg=PA239&lpg=PA239&dq=first+minor+determinant&source=web&ots=BqWTlFMGIB&sig=aeCdnU1sARW9tshE_zhirJZ5dRU&hl=en</ref> == Cofactors and adjugate matrix == The (''i'',''j'') [[Cofactor (linear algebra)|cofactor]] '''C'''<sub>''ij''</sub> of a square matrix '''A''' is just (−1)<sup>''i'' + ''j''</sup> times the corresponding minor '''M'''<sub>''ij''</sub>: :'''C'''<sub>''ij''</sub> = (−1)<sup>''i'' + ''j''</sup> ''M''<sub>''ij''</sub> The [[Cofactor (linear algebra)#Matrix of cofactors|cofactor matrix]] of '''A''', or matrix of '''A''' cofactors, typically denoted '''C''', is defined as the ''n''&times;''n'' matrix whose (''i'',''j'') entry is the (''i'',''j'') cofactor of '''A'''. The [[transpose]] of '''C''' is called the [[adjugate matrix|adjugate]] of '''A'''. Adjugate matrices are used to compute the inverse of square matrices. ==Example== 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 '''C'''<sub>23</sub>. The minor '''M'''<sub>23</sub> 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> where the vertical bars around the matrix indicate that the determinant should be taken. Thus, '''C'''<sub>23</sub> is (-1)<sup>2+3</sup> '''M'''<sub>23</sub> <math> = -13. \!\ </math> ==Applications== The cofactors feature prominently in [[Laplace expansion|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 [[Matrix inversion|inverse]] of small matrices. Given an ''m'' × ''n'' matrix with [[real number|real]] entries (or entries from any other [[field (mathematics)|field]]) and [[rank (matrix theory)|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''']<sub>''I'',''J''</sub> 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''']<sub>''I'',''J''</sub> 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 matrix|Hermitian matrices]], the principal minors can be used to test for [[positive-definite matrix|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 :<math>[\mathbf{AB}]_{I,J} = \sum_{K} [\mathbf{A}]_{I,K} [\mathbf{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. ==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 ''k''th [[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 :<math>\begin{pmatrix} 1 & 4 \\ 3 & \!\!-1 \\ 2 & 1 \\ \end{pmatrix}</math> are &minus;13 (from the first two rows), &minus;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. == References == {{reflist}} {{linear algebra}} [[Category:Matrix theory]] [[Category:Determinants]] [[ar:مختصر (جبر خطي)]] [[cs:Subdeterminant]] [[de:Minor (Mathematik)]] [[fr:Mineur (algèbre linéaire)]] [[he:מינור (אלגברה)]] [[it:Minore (algebra lineare)]] [[lv:Minors (lineārā algebra)]] [[nl:Minor (wiskunde)]] [[pl:Minor macierzy]] [[ru:Минор (линейная алгебра)]]