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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク determinant ： ウィキペディア英語版 determinant In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix. The determinant of a matrix ''A'' is denoted det(''A''), det ''A'', or |''A''|. In the case of a 2 × 2 matrix, the specific formula for the determinant is simply the upper left element times the lower right element, minus the product of the other two elements. Similarly, suppose we have a 3 × 3 matrix ''A'', and we want the specific formula for its determinant |''A''|: :: $=\; aei+bfg+cdh-ceg-bdi-afh\; .$ Each determinant of a 2 × 2 matrix in this equation is called a "minor" of the matrix ''A''. The same sort of procedure can be used to find the determinant of a 4 × 4 matrix, the determinant of a 5 × 5 matrix, and so forth. Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and the determinant can be used to solve those equations, although more efficient techniques are actually used, some of which are determinant-revealing and consist of computationally effective ways of computing the determinant itself. The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, which is essential for eigenvalue problems in linear algebra. In analytical geometry, determinants express the signed n-dimensional volumes of n-dimensional parallelepipeds. Sometimes, determinants are used merely as a compact notation for expressions that would otherwise be unwieldy to write down. It can be proven that any matrix has a unique inverse if its determinant is nonzero. Various other theorems can be proved as well, including that the determinant of a product of matrices is always equal to the product of determinants; and, the determinant of a Hermitian matrix is always real. ==Definition== There are various ways to define the determinant of a square matrix ''A'', i.e. one with the same number of rows and columns. Perhaps the simplest way to express the determinant is by considering the elements in the top row and the respective minors; starting at the left, multiply the element by the minor, then subtract the product of the next element and its minor, and alternate adding and subtracting such products until all elements in the top row have been exhausted. For example, here is the result for a 4 × 4 matrix: :: Another way to define the determinant is expressed in terms of the columns of the matrix. If we write an matrix ''A'' in terms of its column vectors : where the $a\_j$ are vectors of size ''n'', then the determinant of ''A'' is defined so that :$$ \begin & \det \begin a_1, & \ldots, & b a_j + c v, & \ldots, a_n \end = b \det(A) + c \det \begin a_1, & \ldots, & v, & \ldots, a_n \end \\ & \det \begin a_1, & \ldots, & a_j, & a_, & \ldots, a_n \end = - \det \begin a_1, & \ldots, & a_, & a_j, & \ldots, a_n \end \\ & \det(I) = 1 \end where ''b'' and ''c'' are scalars, ''v'' is any vector of size ''n'' and ''I'' is the identity matrix of size ''n''. These equations say that the determinant is a linear function of each column, that interchanging adjacent columns reverses the sign of the determinant, and that the determinant of the identity matrix is 1. These properties mean that the determinant is an alternating multilinear function of the columns that maps the identity matrix to the underlying unit scalar. These suffice to uniquely calculate the determinant of any square matrix. Provided the underlying scalars form a field (more generally, a commutative ring with unity), the definition below shows that such a function exists, and it can be shown to be unique.〔Serge Lang, ''Linear Algebra'', 2nd Edition, Addison-Wesley, 1971, pp 173, 191.〕 Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has ''n'' terms and the coefficient of each product is −1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries. This expression grows rapidly with the size of the matrix (an matrix contributes ''n''! terms), so it will first be given explicitly for the case of matrices and matrices, followed by the rule for arbitrary size matrices, which subsumes these two cases. Assume ''A'' is a square matrix with ''n'' rows and ''n'' columns, so that it can be written as :$$ A = \begin a_ & a_ & \dots & a_ \\ a_ & a_ & \dots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \dots & a_ \end.\, The entries can be numbers or expressions (as happens when the determinant is used to define a characteristic polynomial); the definition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner. The determinant of ''A'' is denoted as det(''A''), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets: : a_ & a_ & \dots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \dots & a_ \end.\, 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「determinant」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.