notes5-1 - λ . The set of all eigenvectors corresponding...

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SECTION 5.1 EIGENVECTORS AND EIGENVALUES SOME DIFFERENTIAL EQUATIONS STUFF. Suppose we have two functions x 1 ( t ) and x 2 ( t ), for example, position and velocity, that model some physical system. Newton or Maxwell or Kirchoff or somebody tells us that these functions satisfy a pair of differential equations like x 0 1 ( t ) = 3 x 1 ( t ) + 2 x 2 ( t ) x 0 2 ( t ) = 4 x 1 ( t ) + 5 x 2 ( t ) . We can easily deal with vectors whose entries are functions, so that we can write x ( t ) = " x 1 ( t ) x 2 ( t ) # , and then x 0 ( t ) = " x 0 1 ( t ) x 0 2 ( t ) # . Now let’s rewrite the system of differential equations using this vector function notation. To see what solutions this system might have, let’s look at a much simpler but analogous case, say y 0 ( t ) = 7 y ( t ) . So, we might expect a solution of our system to have the form x ( t ) = " v 1 v 2 # e rt . For what constants v 1 ,v 2 and r does this happen?
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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
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Unformatted text preview: λ . The set of all eigenvectors corresponding to an eigenvalue λ , together with the zero vector, is called the eigenspace 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 03/06/2010 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas.

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notes5-1 - λ . The set of all eigenvectors corresponding...

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