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Find the condition number of a normal matrices.

My attempt:-

I know condition number of $X\in \mathbb C^{n,n}$ is defined by $\kappa(X)=||X|| \cdot ||X^{-1}||.$ Definition of Normal matrix is given by $X^HX=XX^H.$

We know that $1=||X X^{-1}||\leq ||X|| ||X^{-1}||$ for all consistent norms$^{[\text{Page 5}]}$. Can we prove $||X|| ||X^{-1}||\leq 1$?

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  • $\begingroup$ What if $X = \begin{pmatrix} 1 & 2\\ 2& 1 \end{pmatrix}$ and $\lVert \cdot \rVert$ is the 2-norm? $\endgroup$
    – Hyperbolic PDE friend
    Commented May 24 at 9:39
  • $\begingroup$ You know that $||X|| ||X^{-1}||\geq1$ so that if $||X|| ||X^{-1}||\leq1$ you will have shown that $||X|| ||X^{-1}||=1$. A simple $2\times 2$ example will show you that this can't be the case. $\endgroup$
    – Paul
    Commented May 24 at 9:44
  • $\begingroup$ As some normal matrices are singular, the condition numbers of normal matrices are not bounded above. $\endgroup$
    – user1551
    Commented May 24 at 10:18
  • $\begingroup$ okay. thanks for the comments $\endgroup$
    – Unknown x
    Commented May 24 at 10:26
  • $\begingroup$ What if $X$ is unitary? $\endgroup$
    – Unknown x
    Commented May 24 at 10:28

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