On real and complex S-orthogonal and S-householder matrices - De la Cruz, Ralph John L. (Institute of Mathematics, College of Science, University of the Philippines Diliman, 2014)
Let F=R or C and let S ∈ Mn (F) be nonsingular. An A ∈ Mn (F) is said to be S-orthogonal if A is nonsigular and S-1 A T S = A -1. If in addition, rank (A - 1) =1, we say that A is S-Householder. Assume that S is skew symmetric. It is known that every S-orthogonal matrix is a product of S-Householder matrices. We determine the minimum number of factors needed to write such factorization for both the real and complex case. We determine condition on S such that every S-orthogonal matrix is a product of S-Householder matrices. We also determine conditions on S such that no S-Householder matrix exists.