Transvectant
Encyclopedia
In mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables using Cayley's omega process.
where Ω is Cayley's omega process, the tensor product means take a product of functions with different variables x1,..., xn, and tr means set all the vectors xk equal.
The first transvectant is the Jacobian determinant of the n functions.
The second transvectant is a constant times the completely polarized form of the Hessian
of the n functions.
Definition
If Q1,...,Qn are functions of n variables x = (x1,...,xn) and r ≥ 0 is an integer then the rth transvectant of these functions is a function of n variables given bywhere Ω is Cayley's omega process, the tensor product means take a product of functions with different variables x1,..., xn, and tr means set all the vectors xk equal.
Examples
The zeroth transvectant is the product of the n functions.The first transvectant is the Jacobian determinant of the n functions.
The second transvectant is a constant times the completely polarized form of the Hessian
Hessian matrix
In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
of the n functions.