Similarity invariants for a class of linear controlled systems (Q1874815)

The matrices \(A\), \(B\) and \(C\) defining the linear control system have complex coefficients. Making a linear transformation of the state coordinates we get an algebraic action of the group \(\text{GL}_n(\mathbb{C})\) on the space of linear systems, which is called a similarity action. A theory of invariants leads to classification problems for linear systems. In the present paper the authors suggest a simple proof of the fact that for systems with one-dimensional inputs and outputs the algebra of polynomial \(G\)-invariants of linear systems is free and the basis of invariants consists of algebraically independent elements.