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Unformatted text preview: First Order Linear Systems with Complex Eigenvalues Complex Vectors The space E n (as contrasted with R n ) consists of n-dimensional vectors with complex components: Z = z 1 z 2 . . . z n , where z k = x k + i y k , k = 1 , 2 , ..., n . Clearly we can write Z = X + i Y , where X and Y are the real vectors whose components are the x k and the y k , respectively. Similarly we can define complex matrices: Z = X + iY , where X and Y are real matrices of the same dimension. The rules for multiplying vectors and matrices by scalars, forming linear combinations and multiplying matrices and vectors all remain the same in the complex case. For complex vectors and matrices we have to change the definition of the inner (or dot) product and we have to replace transposition of matrices, as used in the real case, by a modified operation. If Z and W are complex vectors we define Z W = n summationdisplay k =1 z k w k , where w k denotes the conjugate of w k . (We also write W to indicate the conjugate of the complex vector W , the vector whose components are w k .) This modification of the definition of the dot product comes about because, for a complex vector Z , the norm is defined by bardbl Z bardbl = n summationdisplay k =1 | z k | 2 = n summationdisplay k =1 z k z k . Thus, with the new definition of the dot product, we still have Z Z = bardbl Z bardbl 2 . If Z = X + iY is an n m complex matrix, Z * is defined to be the conjugate transpose of Z : Z * = X *- iY * . 1 This reduces to the original transpose operation if Z (= X ) happens to be real. The notation Z T is then used for the simple transpose operation, without conjugation. For complex matrices this operation is very rarely used. So it makes sense to use * in both the real and complex situations, understanding that it means the ordinary transpose in the real case and the conjugate transpose in the complex case. The conjugate transpose matrix Z * is called the adjoint of the complex matrix Z . Consistent with this definition we can also write Z W = W * Z, regarding vectors Z as n 1 matrices. Complex Eigenvalues and Eigenvectors Since a polynomial with real coeffi- cients can have complex roots, a real matrix can have complex eigenvalues. Thus for the matrix M = parenleftbigg 1- k 2 parenrightbigg , with k > 0, we have the characteristic polynomial p ( ) = det parenleftbigg - 1 k 2 parenrightbigg = 2 + k 2 whose roots are i k . What can we say about the eigenvectors? Solving parenleftbigg i k- 1 k 2 i k parenrightbigg parenleftbigg 1 2 parenrightbigg = parenleftbigg parenrightbigg we obtain the eigenvectors = parenleftbigg 1 i k parenrightbigg ....
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