Constructive ways for generating (generalized) real orthogonal matrices as products of (generalized) symmetries

The Cartan–Dieudonné–Scherk (CDS) Theorem of Algebraic Group Theory asserts that for fields of characteristic different from 2, every generalized orthogonal matrix can be written as the product of a certain minimal number of generalized symmetries. When computing over or , these symmetries are called “Householder matrices” in Numerical Linear Algebra. They play an important role in orthogonal elimination and the QR factorization of matrices. This note gives constructive proofs of a number of results to generate a generalized real (or complex) orthogonal (or unitary) matrix as the product of generalized Householder matrices. The proofs are patterned after the standard Householder elimination practises of Numerical Analysis. The original proofs of the CDS Theorem of around 50 years ago were not constructive but relied on the nonexistence of certain isotropic subspaces with respect to the underlying generalized inner product.