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# notes7_2 - MA366 Sathaye Notes on Chapter 7 continued...

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MA366 Sathaye Notes on Chapter 7 continued Continued Notes on early sections of Chapter 7. 1. Complex Eigenvalues(7.6). We now analyze a system DX = AX where A is an n × n matrix with constant (real) coeﬃcients where not all eigenvalues are real. It is useful to concentrate on the case n = 2. Since n = 2 we may assume that the eigenvalues are complex numbers a ± ib . Suppose that for λ = a + ib we ﬁnd an eigenvector p + iq where p and q are the real and imaginary parts of the eigenvector. We then get a part of the solution e at + ibt ( p + iq ) = e at (cos( bt ) + i sin( bt )) ( p + iq ) and this simpliﬁes to e at ( u + iv ) where u = p cos( bt ) - q sin( bt ) and v = p sin( bt ) + q cos( bt ) . Clearly the other eigenvalue a - ib leads to e at ( u - iv ). Combining, we may write the ﬁnal solution as: X ( t ) = c 1 e at u + c 2 e at v. For future reference, we may rewrite this as: X ( t ) = e at ( u v ) C where C = ± c 1 c 2 ² . Finally, we may simplify even further: X ( t ) = e at ( p q ) ± cos( bt ) sin( bt ) - sin( bt ) cos( bt ) ² C. 2. Nature of solutions. (7.5,7.6) Still paying special attention to the n = 2 case and a system DX = X 0 = AX we note the following: Degenerate solutions. The solution X = 0 is a trivial solution curve reducing to a single point, if 0 is not an eigenvalue. If 0 is an eigenvalue, then the line containing its eigenvector is ﬁlled with single point solutions. In the following, we assume that A is non singular or det( A ) 6 = 0. This avoids such degenerate cases, leaving only the origin as a single point solution. We ignore this solution in the rest of the discussion, without further comment! If

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notes7_2 - MA366 Sathaye Notes on Chapter 7 continued...

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