Orthogonal Decomposition Defined by a Pair of Skew-Symmetric Forms
نویسنده
چکیده
Proof. We use induction on n. For n = 1, the result is immediate, First, observe there exist linear transformations Ui (i = 1, 2): X + X such that (U,x, y) = &(x, y). For define&: X -R by Liz(y) = #Q(x, y). Then L,, is a linear functional and since X is an inner product space, there exists Z< E X with L,,(y) = (zi, y) by the canonical isomorphism between X and its dual. Define the transformation U, by zi = Ug; it is easily checked that Ui is linear. The skew-symmetric property of C#Q( *, *) also shows that Ud = Ui*, with Ui* the adjoint of Ui, for (Uix, y) = +i(%, Y) = +i(Ys %) = (UiY> %) = (Y9 U,*X) = (Ui*X, Y).
منابع مشابه
The (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2
Let $Rin textbf{C}^{mtimes m}$ and $Sin textbf{C}^{ntimes n}$ be nontrivial involution matrices; i.e., $R=R^{-1}neq pm~I$ and $S=S^{-1}neq pm~I$. An $mtimes n$ complex matrix $A$ is said to be an $(R, S)$-symmetric ($(R, S)$-skew symmetric) matrix if $RAS =A$ ($ RAS =-A$). The $(R, S)$-symmetric and $(R, S)$-skew symmetric matrices have a number of special properties and widely used in eng...
متن کاملOn Skew-Symmetric Games
By resorting to the vector space structure of finite games, skew-symmetric games (SSGs) are proposed and investigated as a natural subspace of finite games. First of all, for two player games, it is shown that the skew-symmetric games form an orthogonal complement of the symmetric games. Then for a general SSG its linear representation is given, which can be used to verify whether a finite game...
متن کاملProof of a decomposition theorem for symmetric tensors on spaces with constant curvature
In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses – beside the Hodge decomposition for one-forms – an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curv...
متن کاملQuadratic Pairs in Characteristic 2 and the Witt Cancellation Theorem
We define the orthogonal sum of quadratic pairs and we show that there is no Witt cancellation theorem for this operation in characteristic 2. 1. Introduction. Quadratic pairs on central simple algebras were defined in [5]. They play the same role for quadratic forms as involutions for symmetric or skew-symmetric bilinear forms. In particular, they can be used to define twisted orthogonal group...
متن کاملHowe Duality for Lie Superalgebras
We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicity-free decomposition i...
متن کامل