Search Results for ""

Let A=a_(ij) be a matrix with positive coefficients so that a_(ij)>0 for all i,j=1, 2, ..., n, then A has a positive eigenvalue lambda_0, and all its eigenvalues lie on the ...

int_a^b(del f)·ds=f(b)-f(a), where del is the gradient, and the integral is a line integral. It is this relationship which makes the definition of a scalar potential function ...

The determinant G(f_1,f_2,...,f_n)=|intf_1^2dt intf_1f_2dt ... intf_1f_ndt; intf_2f_1dt intf_2^2dt ... intf_2f_ndt; | | ... |; intf_nf_1dt intf_nf_2dt ... intf_n^2dt|.

Given a set V of m vectors (points in R^n), the Gram matrix G is the matrix of all possible inner products of V, i.e., g_(ij)=v_i^(T)v_j. where A^(T) denotes the transpose. ...

Let f_1(x), ..., f_n(x) be real integrable functions over the closed interval [a,b], then the determinant of their integrals satisfies

Let |A| be an n×n determinant with complex (or real) elements a_(ij), then |A|!=0 if |a_(ii)|>sum_(j=1; j!=i)^n|a_(ij)|.

Every complex matrix can be broken into a Hermitian part A_H=1/2(A+A^(H)) (i.e., A_H is a Hermitian matrix) and an antihermitian part A_(AH)=1/2(A-A^(H)) (i.e., A_(AH) is an ...

A theorem proved by É. Cartan in 1913 which classifies the irreducible representations of complex semisimple Lie algebras.

The Hilbert-Schmidt norm of a matrix A is a matrix norm defined by ||A||_(HS)=sqrt(sum_(i,j)a_(ij)^2).

Every modular system has a modular system basis consisting of a finite number of polynomials. Stated another way, for every order n there exists a nonsingular curve with the ...