2 - Elementary Matrix Algebra  pp. 14-44

Elementary Matrix Algebra

By Jer-Nan Juang and Minh Q. Phan

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In the previous chapter, we paid attention to the solution of a scalar ODE. A scalar ODE describes the dynamics of a single-input single output (SISO) dynamic system. In general, a system can have several inputs and several outputs. Such a multi-input multi-output (MIMO) dynamic system can be described by a set of coupled ODEs involving several input and output variables. We must solve these equations simultaneously to obtain the dynamic response of the system. We can find such a solution by using matrix theory, which conveniently rearranges the set of coupled equations in a compact form. Instead of having several coupled scalar ODEs, we now have one single matrix ODE. Matrix operations can be performed on the matrix differential equations, and the final solution can be expressed in a simple form. It is important to realize that applying such matrix operations is equivalent to operating on the scalar ODEs individually, although with the scalar approach it is very easy to miss the general picture. Thus, matrix theory is one fine example in which one gains by simply rewriting an old problem in a new form that can be analyzed more effectively. Another reason why the matrix formulations are so useful is that a set of coupled differential equations of any order can be rewritten as a single matrix differential equation of first order. The same statement applies for a single scalar ODE of any order as well.