C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes5-1 - SECTION 5.1 EIGENVECTORS AND EIGENVALUES EXAMPLES(1 Physical systems are often modeled by a

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SECTION 5.1 EIGENVECTORS AND EIGENVALUES EXAMPLES. (1) Physical systems are often modeled by a state vector v in R p which gives relevant measurements or properties of the system, e.g., temperature, pressure, volume, chemical concentrations, . . . , or temperature, voltage, current, . . . , or population split into age groups or . . . . You can think of others. We can then model the evolution of the system by a transition matrix A , so that if the current state is given by v then the next state is given by A v . Of particular interest would be those steady states for which A v = v , as well as those states for which A v = λ v for some scalar λ . (2) We can easily deal with vectors whose entries are functions, so that we can write x ( t ) = x 1 ( t ) x 2 ( t ) . . . x p ( t ) . Then a system of differential equations with constant coefficients can be written as x 0 ( t ) = A x ( t ) for some matrix A . When would there be a solution of the form x ( t ) =