C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes5-1

C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes5-1 -...

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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 ) = v 1 e rt v 2 e rt . . . v p e rt
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Unformatted text preview: ? IMPORTANT DEFINITIONS. An eigenvector for a p × p matrix A is a nonzero vector v such that A v = λ v for some scalar λ , which is then called an eigenvalue of A . The vector v is called an eigenvector corresponding to the eigenvalue λ . EXAMPLE. Calculate the following and draw some conclusions and make some guesses. 2 1 3-1 5 0 0 -4 5 1 2 1 3-1 5 0 0 1-3 FACT. The eigenvalues of a triangular matrix are the entries on its diagonal. FACT. Eigenvectors that correspond to distinct eigenvalues are linearly independent. WHY? EXAMPLE. Find a basis for the eigenspace corresponding to the eigenvalue λ = 4 for the matrix A = 3 0 2 0 1 3 1 0 0 1 1 0 0 0 0 4 . HOMEWORK: SECTION 5.1...
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This note was uploaded on 04/15/2008 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas.

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C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes5-1 -...

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