tailieunhanh - Handbook of mathematics for engineers and scienteists part 37

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 37', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 220 Algebra 2. Extremal properties of quadratic forms. A point x0 on a smooth surface S is called a stationary point of a differentiable function f defined on S if the derivative of f at the point x0 in any direction on S is equal to zero. The value f xo of the function f at a stationary point x0 is called its stationary value. The unit sphere in a Euclidean space V is the set of all x e V such that x x 1 x 1 . Theorem. Let B x x be a real quadratic form and let B x y Ax y be the corresponding polar bilinear form where A is a Hermitian operator The stationary values of the quadratic form B x x on the unit sphere coincide with eigenvalues of the operator A. These stationary values are attained in particular on the unit eigenvectors ek of the operator A. Remark. If the eigenvalues of the operator A satisfy the inequalities A1 . Xn then A1 and Xn are the largest and the smallest values of B x x on the sphere x x 1. . Second-Order Hypersurfaces 1. Definition of a second-order hypersurface. A second-order hypersurface in an n-dimensional Euclidean space V is the set of all points x 6 V satisfying an equation of the form A x x 2B x c 0 where A x x is a real quadratic form different from identical zero B x is a linear form and c is a real constant. Equation is called the general equation of a second-order hypersurface. Suppose that in some orthonormal basis i1 . in we have A x x XTAX ajXiXk B x BX biXi XT x1 . xn A aij B b1 . bn . Then the general equation of a second-order hypersurface in the Euclidean space V with the given orthonormal basis i1 . in can be written as XT AX 2BX c 0. The term A x x XT AX is called the group of the leading terms of equation and the terms B x c BX c are called the linear part of the equation. . Parallel translation. A parallel translation in a Euclidean space V is a transformation defined by the formulas X X X where X is a fixed point called the new origin. .