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When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. [4]
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [ 1 ] If A is a differentiable map from the real numbers to n × n matrices, then
Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a derivation. That form of the Jacobi identity is also used to define the notion of Leibniz algebra. Another rearrangement shows that the Jacobi identity is equivalent to the following identity between the operators of the adjoint representation:
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
The identity is now obtained by computing (′) in two ways. First, we can directly compute the matrix product ′ (using simple properties of the adjugate matrix, or alternatively using the formula for the expansion of a matrix determinant in terms of a row or a column) to arrive at
Jacobi coordinates, a simplification of coordinates for an n-body system; Jacobi identity for non-associative binary operations; Jacobi's formula for the derivative of the determinant of a matrix; Jacobi triple product, an identity in the theory of theta functions; Jacobi's theorem (disambiguation), several theorems
Jacobi operator (Jacobi matrix), a tridiagonal symmetric matrix appearing in the theory of orthogonal polynomials Topics referred to by the same term This disambiguation page lists mathematics articles associated with the same title.
Thanks to the Jacobi Identity, the three-dimensional cross product gives the structure of a Lie algebra, which is isomorphic to (), the Lie algebra of the 3d rotation group. Because the Jacobi identity fails in seven dimensions, the seven-dimensional cross product does not give R 7 {\displaystyle \mathbb {R} ^{7}} the structure of a Lie algebra.