Figure 1

Matrix Analysis

Bellman has called matrix theory 'the arithmetic of higher mathematics.' Under the influence of Bellman and Kalman, engineers and scientists have found in matrix theory a language for representing and analyzing multivariable systems. Our goal in these notes is to demonstrate the role of matrices in the modeling of physical systems and the power of matrix theory in the analysis and synthesis of such systems.

Beginning with modeling of structures in static equilibrium we focus on the linear nature of the relationship between relevant state variables and express these relationships as simple matrix-vector products. For example, the voltage drops across the resistors in a network are linear combinations of the potentials at each end of each resistor. Similarly, the current through each resistor is assumed to be a linear function of the voltage drop across it. And, finally, at equilibrium, a linear combination (in minus out) of the currents must vanish at every node in the network. In short, the vector of currents is a linear transformation of the vector of voltage drops which is itself a linear transformation of the vector of potentials. A linear transformation of nn numbers into mm numbers is accomplished by multiplying the vector of nn numbers by an mm-by- nn matrix. Once we have learned to spot the ubiquitous matrix-vector product we move on to the analysis of the resulting linear systems of equations. We accomplish this by stretching your knowledge of three-dimensional space. That is, we ask what does it mean that the mm-by- nn matrix XX transforms ℜ n ℜ n (real nn-dimensional space) into ℜ m ℜ m ? We shall visualize this transformation by splitting both ℜ n ℜ n and ℜ m ℜ m each into two smaller spaces between which the given XX behaves in very manageable ways. An understanding of this splitting of the ambient spaces into the so called four fundamental subspaces of XX permits one to answer virtually every question that may arise in the study of structures in static equilibrium.

In the second half of the notes we argue that matrix methods are equally effective in the modeling and analysis of dynamical systems. Although our modeling methodology adapts easily to dynamical problems we shall see, with respect to analysis, that rather than splitting the ambient spaces we shall be better served by splitting XX itself. The process is analogous to decomposing a complicated signal into a sum of simple harmonics oscillating at the natural frequencies of the structure under investigation. For we shall see that (most) matrices may be written as weighted sums of matrices of very special type. The weights are eigenvalues, or natural frequencies, of the matrix while the component matrices are projections composed from simple products of eigenvectors. Our approach to the eigendecomposition of matrices requires a brief exposure to the beautiful field of Complex Variables. This foray has the added benefit of permitting us a more careful study of the Laplace Transform, another fundamental tool in the study of dynamical systems.

--Steve Cox

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This work is licensed by Doug Daniels and Steven J. Cox under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Charlet Reedstrom on Sep 7, 2004 3:18 pm -0500.