This book covers crucial lacunae of the linear discrete-time time-invariant dynamical systems and introduces the reader to their treatment, while functioning under real, natural conditions, in forced regimes with arbitrary initial conditions. It provides novel theoretical tools necessary for the analysis and design of the systems operating in stated conditions. The text completely covers two well-known systems,
IO
and
ISO
, along with a new system,
IIO
. It discovers the concept of the full transfer function matrix
F
(z) in the
z
-complex domain, which incorporates the
Z-
transform of the system, input and another variable, vectors, all with arbitrary initial conditions. Consequently, it addresses the full system matrix
P
(z) and the full block diagram technique based on the use of
(z), which incorporates the Z-transform of the system, input and another variable, vectors, all with arbitrary initial conditions. The book explores the direct relationship between the system full transfer function matrix
(z) and the Lyapunov stability concept, definitions, and conditions, as well as with the
BI
stability concept, definitions, and conditions. The goal of the book is to unify the study and applications of all three classes of the linear discrete-time time-invariant system, for short systems.