mathLinAlgebra5-07-draft

# mathLinAlgebra5-07-draft - ARE211 Fall 2007 LECTURE#15 THU...

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Preliminary draft only: please check for final version ARE211, Fall 2007 LECTURE #15: THU, OCT 18, 2007 PRINT DATE: AUGUST 21, 2007 (LINALGEBRA5) Contents 3. Linear Algebra (cont) 1 3.23. Eigenvalues, eigenvectors and difference Equations 1 3. Linear Algebra (cont) 3.23. Eigenvalues, eigenvectors and difference Equations Tremendously important application of eigenvalues and eigenvectors relates to difference equations. Consider a system of linear homogeneous first-order difference equations of the form x t = A x t - 1 . Think of this as an infinite system of equations, one for each time period. Each initial vector generates a sequence of vectors { x 0 , x 1 . . . } s.t. t > 0 , x t = A x t - 1 : any such sequence is called a solution to the system what are the properties of these solutions: do they converge to some specific vector? A steady state/equilibrium/stationary solution to an equation system is a vector x satisfying x = A x . Familiar result is that given a system of difference equations of the form x t = A x t - 1 such

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