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I couldn't find any answer how to explain whether $A$ matrix and $A^{-1}$ have the same condition number or not.

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The condition number of an invertible matrix $A$ is $\|A\|\|A^{-1}\|$ and so the condition number of $A^{-1}$ is $\|A^{-1}\|\|(A^{-1})^{-1}\| = \|A^{-1}\|\|A\|$. Hence, $A$ and $A^{-1}$ have the same condition number.

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  • $\begingroup$ Isn't that only true (with the generally accepted meaning of the condition number of a matrix) if $\|A\|$ is the spectral norm of $A$? Maybe, you should mention that in your answer. $\endgroup$
    – Tobias
    Jun 23, 2020 at 13:36
  • $\begingroup$ The definition of condition number in my answer applies to any consistent matrix norm $\|\cdot\|$ and is not restricted to the spectral norm. $\endgroup$
    – K. Miller
    Jun 23, 2020 at 23:12
  • $\begingroup$ Okay, you are right. The statement includes that one for the condition number on the basis of the spectral norm which is AFAIK the most often used one for estimating the sensitivity of numerical algorithms for matrices. $\endgroup$
    – Tobias
    Jun 24, 2020 at 7:03

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