tailieunhanh - Handbook of mathematics for engineers and scienteists part 36

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 36', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Bilinear and Quadratic Forms 213 . Canonical form of linear operators. An element x is called an associated vector of an operator A corresponding to its eigenvalue A if for some m 1 we have A - AI x 0 A - AI 1 x 0. The number m is called the order of the associated vector x. Theorem. Let A bea linear operator in an n-dimensional unitary space V. Then there is a basis i k 1 2 . l m 1 2 . nk n1 n2 nl n in V consisting of eigenvectors and associated vectors of the operator A such that the action of the operator A is determined by the relations Aik Ak ik k 1 2 . l Ai Aki i -1 k 1 2 . l m 2 3 . nk . Remark 1. The vectors ii. k 1 are eigenvectors of the operator A corresponding to the eigenvalues Xk. Remark 2. The matrix A of the linear operator A in the basis i has canonical Jordan form and the above theorem is also called the theorem on the reduction of a matrix to canonical Jordan form. . Bilinear and Quadratic Forms . Linear and Sesquilinear Forms . Linear forms in a unitary space. A linear form or linear functional on V is a linear operator A in L V C where C is the complex plane. Theorem. For any linear form f in a finite-dimensional unitary space V there is a unique element h in V such that f x x h forall x e V. Remark. This statement is true also for a Euclidean space V and a real-valued linear functional. . Sesquilinear forms in unitary space. A sesquilinear form on a unitary space V is a complex-valued function B x y of two arguments x ye V such that for any x y z in V and any complex scalar A the following relations hold 1. B x y z B x z B y z . 2. B x y z B x y B x z . 3. B Ax y AB x y . 4. B x Ay AB x y . 214 Algebra Remark. Thus B x y is a scalar function that is linear with respect to its first argument and antilinear with respect to its second argument. For a real space V sesquilinear forms turn into bilinear forms see Paragraph . Theorem. Let B x y be a sesquilinear form in a unitary space V. Then there is a .