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Lecture2B - 1 Chapter 2 part B 2.5 Complex Eigenvalues Real...

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1 Chapter 2 part B 2.5 Complex Eigenvalues Real Canonical Form A semisimple matrix with complex conjugate eigenvalues can be diagonalized using the procedure previously described. However, the eigenvectors corresponding to the conjugate eigenvalues are themselves complex conjugate and the calculations involve working in complex n-dimensional space. There is nothing wrong with this in principle, however the manipulations may be a bit messy. Example : Diagonalize the matrix . Eigenvalues are roots of the characteristic polynomial. . . The eigenvalues are and . Eigenvectors are solutions of . Obtain and . Then from we need to compute . The transformation matrix . Computing requires care since we have to do matrix multiplication and complex arithmetic at the same time. If we now want to solve an initial value problem for a linear system involving the matrix , we have to compute and . This matrix product is pretty messy to compute by hand. Even using a symbolic algebra system, we may have to do some work to convert our answer for into real form. Carry out the matrix product in Mathematica instead using ComplexDiagonalization1.nb. Discuss the commands Eigenvalues, Eigenvectors, notation for parts of expressions, Transpose, MatrixForm, Inverse and the notation for matrix multiplication. Obtain and . Alternatively, there is the Real Canonical Form that allows us to stay in the real number system. Suppose has eigenvalue , eigenvector and their complex conjugates. Then writing in real and imaginary parts: Taking real and imaginary parts
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2 Chapter 2 part B Consider the transformation matrix . These equation can be written . The exponential of the block on the right was computed at the end of section 2.3 (Meiss, Eq. 2.31). Example . Let . Find its real canonical form and compute . We have already found the eigenvalues and eigenvectors. Setting we have , . The transformation matrix and its inverse are , . Find , . Using Meiss 2.31 . Compute . Find , . Diagonalizing an arbitrary semisimple matrix
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3 Chapter 2 part B Suppose has real eigenvalues and pairs of complex conjugate ones. Let be the corresponding real eigenvectors and , be the real and imaginary parts of the complex conjugate eigenvectors. The transformation matrix is nonsingular and where . The solution of the initial value problem will involve the matrix exponential . In this way we compute the matrix exponential of any matrix that is diagonalizable. 2.6 Multiple Eigenvalues The commutator of and is . If the commutator is zero then and commute. Fact. If and , then . Proof . . □ Generalized Eigenspaces Let where . Recall that eigenvalue and eigenvector satisfy . This can be rewritten as . Suppose has algebraic multiplicity 1. Then the associated eigenspace is
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4 Chapter 2 part B . A space is invariant under the action of if implies . For example, is invariant under by the fact above.
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