{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online