A constructive proof of the Cartan-Dieudonné-Scherk theorem in the real or complex case
Let \(\mathbb{F}\) be a field with char\((\mathbb{F)}\neq2\), and \(S\) be an \(n\times n\) nonsingular symmetric matrix over \(\mathbb{F}\). An \(S\)-Householder matrix is an \(n\times n\) matrix of the form \(H=I-2uu^{T}S/(u^{T}Su)\) for a vector \(u\) such that \(u^{T}Su\neq0\). An \(n\times n\) matrix is \(S\)-orthogonal if \(Q^{T}SQ=S\) (so each \(S\)-Householder matrix is \(S\)-orthogonal). The Cartan-Dieudonné-Scherk (CDS) theorem states that every \(S\)-orthogonal matrix \(Q\) is a product of at most \(k\) \(S\)-Householder matrices where \(k=\text{rank}(Q-I)\) if \(S(Q-I)\) is not skew symmetric, and \(k=\text{rank}(Q-I)+2\) if \(S(Q-I)\) is skew symmetric. This value of \(k\) is best possible. There are many proofs of the CDS theorem, and some constructions which provide an explicit factorization into \(S\)-Householder matrices but using more than the minimal number of factors. In the case that \(\mathbb{F=R}\) or \(\mathbb{C}\) the author describes a constructive process to factor an \(S\)-orthogonal matrix \(Q\) into \(k\) \(S\)-Householder matrices where \(k\) is the CDS bound.
scientific article; zbMATH DE number 5853399