Abstract
In this chapter, we describe the foundations of the theory of non-degenerate symmetric bilinear forms on finite-dimensional vector spaces and their orthogonal groups. Among the highlights of this discussion are the Cartan–Dieudonné Theorem, which states that any orthogonal transformation is a finite product of reflections, and Witt’s Theorem giving a partial normal form for quadratic forms. The theory of split symmetric bilinear forms is found to have many parallels to the theory of symplectic forms, and we will give a discussion of the Lagrangian Grassmannian in this spirit.
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Meinrenken, E. (2013). Symmetric bilinear forms. In: Clifford Algebras and Lie Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36216-3_1
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DOI: https://doi.org/10.1007/978-3-642-36216-3_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36215-6
Online ISBN: 978-3-642-36216-3
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