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Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$. For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} A S=D }\}$, then $K(A)=\min_{G_A}\{{\|S\| \|S^{-1}\|}\}$ is called the spectral condition number of A. Can we bound the spectral condition number $K(A)$ of any diagonalizable matrix A?

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See: An upper bound for the spectral condition number of a diagonalizable matrix (1997)

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  • $\begingroup$ I have read this paper. But unfortunately, this bound looks too relaxed. The bound is exponential on the order of the matrix. $\endgroup$ Jan 25, 2021 at 9:46
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    $\begingroup$ this may well be the best you can do without further info on the matrix; if the matrix is diagonally dominant, you can do better, see math.stackexchange.com/q/3520012/87355 $\endgroup$ Jan 25, 2021 at 9:54
  • $\begingroup$ math.stackexchange.com/q/3520012/87355 handles the condition number, not the spectral condition number. How can one obtain a bound on $K(A)$ from math.stackexchange.com/q/3520012/87355 knowing that $A$ is diagonally dominant? (Actually I'm also interested in the case where $A$ is tridiagonal) $\endgroup$
    – bas
    Feb 8, 2023 at 8:39

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