Implicit Linear Algebra and its Applications

Abstract

Linear systems often involve, as a basic building block, solutions of equations of the form \begin{align*} A_Sx_S&+A_Px_P =0\\ A'_Sx_S & =0, \end{align*} where our primary interest might be in the vector variable $x_P.$ Usually, neither $x_S$ nor $x_P$ can be written as a function of the other but they are linked through the linear relationship, that of $(x_S,x_P) $ belonging to $\mathcal{V}_{SP},$ the solution space of the first of the two equations. If $\mathcal{V}_{S}$ is the solution space of the second equation, we may regard the final space of solutions $\mathcal{V}_{P}$ as derived from the other two spaces by an operation, say, `$\mathcal{V}_{P}=\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_{S}.$' This operation, together with linear relationships, can be used to build a version of linear algebra which we call `implicit linear algebra'. There are two basic results - an `implicit inversion theorem' which describes when $\mathcal{V}_{S}$ can be obtained from $\mathcal{V}_{P}$ and $\mathcal{V}_{SP},$ and an `implicit duality theorem' which says $\mathcal{V}_{P}^{\bot}=\mathcal{V}_{SP}^{\bot}\leftrightarrow \mathcal{V}_{S}^{\bot}.$ These notions originally arose in the building of circuit simulators. They have been reinterpreted for the present purpose. Using them, we develop an algorithmic version of linear multivariable control theory, avoiding the computationally expensive idea of state equations. We replace them by `emulators', which are easy to build, but can achieve most of whatever can be done with state equations. We define the notions of generalized autonomous systems and generalized operators, and develop a primitive spectral theory for the latter. Using these ideas, we develop the usual controllability - observability duality, state and output feedback, pole placement etc.