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Given an $n \times n$ complex matrix A. Find sequences of matrices ${S_i}, {D_i}$, such that $S_i D_i {S_i}^{-1}$ converges to $A$. Where the $D_i$ are diagonal with distinct eigenvalues, and the $S_i$ are invertible. Convergence means convergence in each entry.

I thought about using the fact that the set of diagonalizable matrices is dense in the set of all complex matrices. But I'd need a stronger fact that the set of diagonalizable matrices with distinct eigenvalues are dense in the set of complex matrices. Which is apparently true since it's exactly what the problem is asking for.

Is there an explicit construction? Or an easy way to piggy-back off the proof of diagonalizable matrices are dense in complex matrices?

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You mean diagonalizable matrices are dense, not diagonal matrices. –  Robert Israel Sep 19 '12 at 18:57
    
Yes, I do. My bad. –  Stuart Sep 19 '12 at 18:58

1 Answer 1

Hint: given an upper triangular matrix, making small changes to the diagonal elements can give you a matrix with all distinct eigenvalues.

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