L96 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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Fall 2003 Math 308/501–502 9 Matrix Methods for Linear Systems 9.6 Complex Eigenvalues Wed, 19/Nov c ± 2003, Art Belmonte Summary Let A be an n × n real matrix and let p ( r ) be its characteristic polynomial. The Fundamental Theorem of Algebra guarantees that p ( r ) factors as ( - 1 ) n k Y j = 1 ( r - r j ) q j ,where r 1 ,... r k are the distinct eigenvalues of A and k X j = 1 q j = n . Definitions The algebraic multiplicity of r j is q j ; i.e., the number of times r - r j appears in the factorization of p ( r ) . The geometric multiplicity of r j is d j , dimension of the eigenspace of r j ; i.e., the dimension of nullspace of A - r j I . We always have 1 d j q j . If we’re lucky, we have d j = q j for j = 1 ,..., k . For in this case we have k X j = 1 d j = k X j = 1 q j = n and thus a full set of linearly independent eigenvectors from which to construct a fundamental solution set. (If we’re not lucky, we resort to the Jordan canonical form.) Facts If r is an eigenvalue of A with associated eigenvector v ,then x = e rt v is a solution of x 0 = Ax . (If r is real, then x is real-valued. If r is complex, then x is complex-valued.) If r 1 r k , are distinct eigenvalues of A with associated eigenvectors v 1 v k , then these eigenvectors are linearly independent. If A has n distinct eigenvalues r 1 r n , with associated eigenvectors v 1 v n x k ( t ) = e r k t v k ,forma fundamental set of solutions for the system x 0 = Ax . Suppose that r 1 , r 2 = α ² β i (where β> 0) is a pair of complex conjugate eigenvalues of A .Le t w = a + i b be a complex eigenvector associated with the eigenvalue r 1 = α + β i .(Here a and b are the real and imaginary parts of w , respectively.) Then a pair of real solutions of x 0 = Ax
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This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

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L96 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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