Orthogonalization, factorization, and identification as to the theory of recursive equations in linear algebra

We outline theoretical foundations for the recurrent algorithms of computational linear algebra based on counter orthogonalization processes over an ordered system of vectors; we also show the importance of these processes for analysis and applications. We present some important applications of counter orthogonalization processes related to some approximation problems and signal processing as well as recent applications related to the so called homogeneous structures and Toeplitz systems. In particular, these applications contain operators and inversion of matrices, $\mathbb{QDR}$- and $\mathbb{QDL}$-decompositions, $\mathbb{RDL}$- and $\mathbb{LDR}$-factorizations, solutions of integral equations and of systems of algebraic equations, signal estimation on based on approximation models in the form of differential and difference equations and variational identification (coefficients estimation) of the latter.