Original matrix formulas obtained by differentiation of the system matrix in respect to movements of components are
derived. Components kinematics for the three zoom systems realized by means of interactive graphical software is presented. An optical system may be structurally designed by successive steps and its parameters determined to fulfil
requirements, such as optical conjugation, focal lengths or magnifications. Improved software developed in this work
serves both determination of optical powers and separations and movements of components. Developed methodology covers different types of fixed and zoom systems, the latter type with electronic or optical compensation. One may consider any optical system, such as the reproduction lens, objective lens or telescope system, because matrix optics distinguishes them remarkably easy. Kinematics pertaining to a full tract of the zoom system is determined at a discrete number of positions. Movements of so-called basic variable components are determined in a full cycle of work by means of iterative methods while movements of supplementary components may be inserted by means of exponential-parabolic functions also including their linear form. Any component of the zoom system may act as a variable, supplementary or fixed component, but it is mainly dependent on the structural design. Parameters of characteristics are computed as elements of a certain matrix. Designing is that to set these elements on required values by means of system parameters or movements of components. In this way, one may create complex multi-group systems with characteristics and movements which we accept. Properties of these systems are presented by numerical and graphical forms. Advantages of these systems are their more compact construction, more smooth kinematics, and better possibilities of optimization, what is particularly valuable for zoom systems with a high zooming ratio.