Given a real nxn matrix A and a perturbation of the form A+R where R is chosen uniformly at random in [-r,r]nxn i.e. each entry of R is in the segment [-r,r] for some given r>0, what is the probability that A+R would be real diagonalizable ? i.e. what is the probability for the existance of real invertible matrix T such that T^{-1}(A+B)T would have the form of block diagonal with real blocks of sizes 1x1 or 2x2 only ?
Another related (but not equivalent) question: what is the probablility that all the eigenvalues of A+R would be different (and therefore, A+R would be real diagonalizable) ?
Dear Yossi,
I suggest you to see links and attached files in subject.
-Lecture 30: Diagonalization - Harvard Math Department
www.math.harvard.edu/~knill/teaching/math19b.../lecture30.pdf
-The Eigenvalue Problem: Perturbation Theory
www.math.usm.edu/lambers/mat610/sum10/lecture13.pdf
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Best regards
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