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Determinant - Applications | | Determinant - Applications: Encyclopedia II - Determinant - Applications | | Determinants are used to characterize invertible matrices (namely as those matrices, and only those matrices, with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule. They can be used to find the eigenvalues of the matrix A through the characteristic polynomial
where I is the ide ...
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 - Applications
Determinant - Applications
Determinants are used to characterize invertible matrices (namely as those matrices, and only those matrices, with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule. They can be used to find the eigenvalues of the matrix A through the characteristic polynomial
where I is the identity matrix of the same format as A.
One often thinks of the determinant as assigning a number to every sequence of n vectors in , by using the square matrix whose columns are the given vectors. With this understanding, the sign of the determinant of a basis can be used to define the notion of orientation in Euclidean spaces. The determinant of a set of vectors is positive if the vectors form a right-handed coordinate system, and negative if left-handed.
Determinants are used to calculate volumes in vector calculus: the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if the linear map is represented by the matrix A, and S is any measurable subset of , then the volume of f(S) is given by . More generally, if the linear map is represented by the m-by-n matrix A, and S is any measurable subset of , then the n-dimensional volume of f(S) is given by . By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines.
The volume of any tetrahedron, given its vertices a, b, c, and d, is (1/6)·|det(a−b, b−c, c−d)|, or any other combination of pairs of vertices that form a simply connected graph.
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 "Applications", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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