By Joachim Hilgert

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D are the real eigenvalues of x, then eλ1 , . . , eλd are the eigenvalues of ex . Therefore ex is positive definite for each hermitian matrix x. If, conversely, g ∈ Pdd (K), then let v1 , . . , vd be an orthonormal basis of eigenvectors for g with gvj = λj vj . 3 The Logarithm Function 47 logH (g) ∈ Hermd (K) by logH (g)vj := (log λj )vj , j = 1, . . , d. From this construction of the logarithm function it is clear that logH ◦ expP = idHermd (K) and expP ◦ logH = idPdd (K) . For two real numbers x, y > 0, we have log(xy) = log x + log y.

Ii) If β is nondegenerate and ψ ∈ GL(V ) commutes with all symplectic transvections, then every vector in v is an eigenvector of ψ. 16. Let V be a finite-dimensional K-vector space for K = R or C and and β be a non-degenerate symmetric bilinear form on V . An involution ϕ ∈ O(V, β) is called an orthogonal reflection if dimK im (ϕ−idV ) = 1. Show that: (i) For each orthogonal reflection ϕ, there exists a non-isotropic vϕ ∈ V such β(v,vα ) that ϕ(v) = v − 2 β(v . α ,vα ) (ii) If ψ ∈ GL(V ) commutes with all orthogonal reflections, then every nonisotropic vector for β is an eigenvector of ψ, and this implies that ψ ∈ K× idV .

In view of Φ(1) = idV and Φ(AB) = Φ(A)Φ(B), we obtain a group isomorphism Φ|GLn (K) : GLn (K) → GL(V ). (b) Let V be an n-dimensional vector space with basis v1 , . . , vn and β : V × V → K a bilinear map. ,n is an (n × n)-matrix, but this matrix should NOT be interpreted as the matrix of a linear map. It is the matrix of a bilinear map to K, which is something different. , an element of K. We write Aut(V, β) := {g ∈ GL(V ) : (∀v, w ∈ V ) β(gv, gw) = β(v, w)} for the isometry group of the bilinear form β.

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Structure and Geometry of Lie Groups by Joachim Hilgert


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