A new point of view for the householder matrix by using matrix exponential

Abstract

It is well known that many approaches can yield an orthogonal matrix. Householder employed the mirror technique to derive a symmetric orthogonal matrix. The Householder matrices of odd orders are found to have the matrix forms of e At and e iBt , and those of even orders are found to have the matrix forms of e iBt , respectively, where A is an anti-symmetric matrix, B is a symmetric matrix and t is a specified time. Householder matrices with dimension of one by one to five by five are constructed by the proposed formulation. Also, the relations among orthogonal, Householder and Hermitian matrices are discussed.