Norms of principal axis realizations: Application of abstract realization theory

Principal axis realizations are an important class of implementations, associated with good numerical properties. The spectral (l₂) norm of the principal axis realization matrices of arbitrary transfer functions is studied. In particular, it is shown that if the transfer function is contractive, then its principal axis realization matrices have a norm less than √2, while the internally balanced realization is conjectured to have a norm of less than 1. Furthermore, if the transfer function is an all-pass transfer function, then the three principal axis realizations become identical and equal to an orthogonal matrix. Abstract realization theory is used to prove these results