Thursday, March 5, 2009
Prize problem
In the draft version of the lecture notes chapter 5 I suggest in the proof of part (b) of Proposition 5.1.2 that it is possible to estimate the eigenvalues of a matrix (A+B) as a function of the eigenvalues of the matrix A and the norm |B| of matrix B. In fact, I meant to ask you to work this out in Exercise sheet 5, question 1(a). The question is posed slightly less specific there, and on the solution sheet I answer the question without giving an explicit estimate to the change of the eigenvalues of (A+B) as a function of |B|, only arguing that as |B| goes to zero, the eigenvalues of (A+B) converge to those of A.
Prize question: Let A be a real mxm matrix with set of eigenvalues lambda_1,...,lambda_m with real part >= epsilon>0. Find some explicit bound for delta (as a function of epsilon, where delta(epsilon)>0 if epsilon>0) such that the eigenvalues of (A+B) are guaranteed to not intersect the imaginary axis for all B with |B| smaller than delta(epsilon).
Please send your answer(s) to me by e-mail before March 23. A box of chocolates will be made available to the author(s) with the - in my view - best answer (if quality is equal the answer having been sent-in first will prevail).
Prize question: Let A be a real mxm matrix with set of eigenvalues lambda_1,...,lambda_m with real part >= epsilon>0. Find some explicit bound for delta (as a function of epsilon, where delta(epsilon)>0 if epsilon>0) such that the eigenvalues of (A+B) are guaranteed to not intersect the imaginary axis for all B with |B| smaller than delta(epsilon).
Please send your answer(s) to me by e-mail before March 23. A box of chocolates will be made available to the author(s) with the - in my view - best answer (if quality is equal the answer having been sent-in first will prevail).
# posted by Jeroen Lamb @ 11:07 AM 
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