Inside Collection (Course): Matrix Analysis

Summary: This modules lays out the structure for the text of the CAAM 335 course in matrix analysis.

Matrix Analysis |
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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 *visualize * this transformation by
splitting both

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

--Steve Cox