ABSTRACT
In many branches of engineering and science it is desirable to be able to mathematically determine the state of a system based on a set of physical relationships. These physical relationships may be determined from characteristics such as circuit topology, mass, weight, or force, to name a few. For example, the injected currents, network topology, and branch impedances govern the voltages at each node of a circuit. In many cases, the relationship between the known, or input, quantities and the unknown, or output, states is a linear relationship. Therefore, a linear system may be generically modeled as
Ax = b (2.1)
where b is the n× 1 vector of known quantities, x is the n× 1 unknown state vector, and A is the n×n matrix that relates x to b. For the time being, it will be assumed that the matrix A is invertible, or non-singular; thus each vector b will yield a unique corresponding vector x. Thus the matrix A−1 exists and
x∗ = A−1b (2.2)
is the unique solution to Equation (2.1).
