Orthogonal matrix decompositions in systems and control

In this paper we present several types of orthogonal matrix decompositions used in systems and control. We focus on those related to eigenvalue and singular value problems and include generalizations to several matrices. 1. Introduction In systems and control theory, one often uses state space models to represent a dynamical system. In such models the relation between inputs u(t) 2 R m and outputs y(t) 2 R p is described via the use of a state x(t) 2 R n and a system of first order differential equations _ ae E x(t) = Ax(t) +Bu(t) y(t) = Cx(t) +Du(t); (1) where E and A are real n Theta n matrices and B, C and D are real n Theta m, p Theta n and p Theta m matrices, respectively. In the above model (1) the input, output and state vectors are continuous time functions. An analogous model is used for discrete time vectors functions u k , y k and x k , now involving a system of first order difference equations _ ae Ex k+1 = Ax k + Bu k y k = Cx k +Du k _ (2)