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Unformatted text preview: Notes for Day Review : . : Complex Eigenvalues Last time, we discussed the general solution to a homogeneous system of rst-order equations where the eigenpairs are real and distinct: If the eigenvalues are , , solution is: , n , and the corresponding eigenvectors are e , e , , en , respectively, then the general y = c e te + c e te + + c n e n t en . is formula applies whenever all eigenvalues are real and distinct, and also when there are repeated eigenvalues which correspond to linearly dependent eigenvectors. is time, we'll discuss what happens if there are complex eigenvalues: Conjugate pairs
Because complex eigenvalues are solutions to a characteristic equation with real coe cients, complex eigenvalues always occur in conjugate pairs. So, for example, if + i is an eigenvalue, then - i must also be an eigenvalue. Complex eigenvalues have complex eigenvectors, and these also occur in conjugate pairs. For example, if the eigenvector corresponding to + i is i , then the eigenvector corresponding to - i is - i . In other words, if you have already computed an eigenvector for a complex eigenvalue, then you do not need to compute an eigenvector for its conjugate "from scratch". Turning Complex Solutions into Real Solutions
ere are complex-valued functions that solve di erential equations, and they have the form (following the previous discussion) y(t) = e ( system.
+ )t i but the Existence and Uniqueness theorem guarantees a real-valued solution to the ): Turning the complex solution into a real solution relies on Euler's Formula (Equation , p. e i t = cos t + i sin t Using Euler's Formula with these conjugate pairs of eigenvalues and eigenvectors will give us a real part of the solution, and an imaginary part of the solution. e ( nal) imaginary part that we obtain (but not the intermediate steps) will turn out to still be a solution even if we drop the i. (We saw an example like this in . .) p. , Example
- . Find a real-valued fundamental set of solutions of y = Ay, where A = p. , Example
- - - , eigenpairs are: - i x For the system y = Ay, with A = + i - i Find the general solution to the system. -i ...
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This note was uploaded on 04/05/2008 for the course MATH 2214 taught by Professor Edesturler during the Spring '06 term at Virginia Tech.
- Spring '06