notes-jury - Notes on the Jury conditions Patrick De...

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Unformatted text preview: Notes on the Jury conditions Patrick De Leenheer * January 16, 2009 Here we will solve problem 2.4.15 which states that it is necessary and sufficient for a real 2 by 2 matrix J to have eigenvalues with modulus less than 1, that: | tr J | < 1 + det J < 2 , (1) where tr J is the trace of J (i.e. the sum of the diagonal entries of J ), and det J is the determinant of J . The practical relevance of these so-called Jury conditions is that instead of calculating the eigenvalues of J , and checking if their modulus is less than 1, we can verify this with the above inequalities which are expressed in quantities (trace and determinant) that are easily calculated, once J is given. Proof. Setting J = j 11 j 12 j 21 j 22 , we see that the characteristic equation det( J- I 2 ) = 0 is 2- ( j 11 + j 22 ) +( j 11 j 22- j 12 j 21 ) = 0, which in short is p ( ) = 2- tr J + det J = 0 . (2) Lets show that (1) are necessary conditions, but notice first that (1) is the same as the following...
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