Let A $\epsilon$ $R^{n*n}$ be invertible and $1\leq p,q\leq \infty $ with $1/p+1/q=1.$ Show that $k_p(A)=k_q(A^T)$. To be clear $k_p(A)$ is the relative condition number with the Lp norm. Similarly $k_q(A^T)$ is the relative condition number with the Lq norm. I am not sure exactly where to go but I started by transforming the equation into the definition of the relative condition factor to get: $||A||_p||A^{-1}||_p=||A^T||_q||{A^T}^{-1}||_q $.

  • $\begingroup$ I am working on this exact question now and still haven't figured it out. Hoping someone can elaborate! This question appears in Wendland's Numerical Linear Algebra and, to be honest, I am confused as all heck with it. $\endgroup$ – Carah Sep 12 '19 at 19:16

$$\|A\|_p = \sup_{\|x\|_p = 1} \|A x\|_p = \sup_{\|x\|_p = 1} \sup_{\|y\|_q = 1} |y^T A x|$$ and similarly for $\|A^T\|_q$. Same with $A$ replaced by $A^{-1}$.

  • $\begingroup$ Would it be possible for you to elaborate at all? In the textbook I'm using where this exact question appears, there are no theorems/definitions of $||A||_p$ that involve two sup's... I'm pretty confused. Thanks for your input. $\endgroup$ – Carah Sep 12 '19 at 19:22
  • $\begingroup$ See Hölder's inequality, in particular the case of counting measure $\endgroup$ – Robert Israel Sep 13 '19 at 2:01

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