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Determinant - Generalizations and related functions | | Determinant - Generalizations and related functions: Encyclopedia II - Determinant - Generalizations and related functions | | | As was pointed out above, it is possible to unambiguously define the determinant of any linear map f : V → V, if V is a finite-dimensional vector space.
It makes sense to define the determinant for matrices whose entries come from any commutative ring. The computation rules, the Leibniz formula and the compatibility with matrix multiplication remain valid, except that now a matrix A is invertible if and only if See also: Determinant, Determinant - Determinants of 2-by-2 matrices, Determinant - Applications, Determinant - General definition and computation, Determinant - Example, Determinant - Properties, Determinant - Derivative, Determinant - Generalizations and related functions, Determinant - Algorithmic implementation, Determinant - History | | | Determinant, Determinant - Algorithmic implementation, Determinant - Applications, Determinant - Derivative, Determinant - Determinants of 2-by-2 matrices, Determinant - Example, Determinant - General definition and computation, Determinant - Generalizations and related functions, Determinant - History, Determinant - Properties | | |
| | | Determinant: Encyclopedia II - Determinant - Generalizations and related functions
Determinant - Generalizations and related functions
As was pointed out above, it is possible to unambiguously define the determinant of any linear map f : V → V, if V is a finite-dimensional vector space.
It makes sense to define the determinant for matrices whose entries come from any commutative ring. The computation rules, the Leibniz formula and the compatibility with matrix multiplication remain valid, except that now a matrix A is invertible if and only if det(A) is an invertible element of the ground ring.
Abstractly, one may define the determinant as a certain anti-symmetric multilinear map as follows: if R is a commutative ring and M = Rn denotes the free R-module with n generators, then
is the unique map with the following properties:
- det is R-linear in each of the n arguments.
- det is anti-symmetric, meaning that if two of the n arguments are equal, then the determinant is zero.
- , where ei is that element of M which has a 1 in the i-th coordinate and zeros elsewhere.
Linear algebraists prefer to use the multilinear map approach to define determinant, whereas combinatorialists may prefer the Leibniz formula. (Of course, even when using the above abstract approach, one has to use the Leibniz formula to show that such a multilinear map actually exists.)
The Pfaffian is an analog of the determinant for antisymmetric matrices. It is a polynomial of degree n, and its square is equal to the determinant of the matrix.
There is no direct generalisation of determinants, or of the notion of volume, to spaces of infinite dimension. There are various approaches possible, including the use of the extension of the trace of a matrix, and functional determinants.
Other related archives16th century, Bezout, Binet, Cardano, Catalan, Cauchy, Cauchy-Binet formula, Cayley, Cholesky decomposition, Christoffel, Cramer, Cramer's rule, Crelle, Euclidean spaces, Frobenius, Gauss, Gauss algorithm, Glaisher, Gottfried Leibniz, Hankel, Hesse, Hessians, Jacobi, Jacobi's formula, Jacobian, Jordan normal form, LU decomposition, Lagrange, Laplace, Laplace's formula, Lebesgue, Leibniz, Muir, Pfaffian, Pfaffians, Scott, Sylvester, Vandermonde, Wronskians, absolute value, adjugate, area, basis, calculus, characteristic polynomial, circulants, cofactors, commutative ring, complex, complex numbers, conjugate, coordinate system, differentiable, dimensional, discriminant, eigenvalues, elimination theory, even and odd permutations, exponential function, factorial, field, free R-module, function, functional determinants, graph, identity matrix, invertible matrices, linear algebra, linear equations, linear map, linear transformation, matrix multiplication, matrix norms, measurable, minor, minors, multilinear algebra, multilinear map, of order, orientation, orthogonal transformation, parallelepiped, parallelogram, permutations, polynomial function, positive, quantic, real, ring, scalar, scalars, scale factor, sequence, signature, similar, skew lines, square matrix, square root, subset, substitution rule, system of linear equations, tetrahedron, trace function, trace of a matrix, transformation matrix, transpose, triangular matrix, unit, unit square, vector calculus, vector space, volume, volumes
 Adapted from the Wikipedia article "Generalizations and related functions", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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