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For $n\times n$ matrix $A$, $||A||_F^2=\sum_{ij}a_{ij}^2$. Can we show that $||CXC^{-1}||_F^2\geq \sum_i x_i^2$, where $C$ is invertible matrix, $X=diag(x_1,\cdots,x_n)$ is an diagonal matrix with diagonals $x_1,\cdots,x_n$?

If $C$ is just the identity matrix, it is OK.

xldd
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  • You asked a very similar question yesterday and I gave you an answer that did not really apply with the $\circ$ operation you defined then. However, here you can show that $|A|_F^2 = \mathrm{tr}(AA^\intercal).$ Then, $|CXC^{-1}|_F^2 = \mathrm{tr}(CXC^{-1}(C^{-1})^\intercal X^\intercal C^\intercal).$ If $C$ was orthogonal, then $C^{-1} = C^\intercal$ and you could cancel out all the $C$ and $C^\intercal.$ This would yield an equality. Why do you think you can get the lower bound? – William M. Aug 12 '21 at 14:51
  • Notice that $AX$ means a new matrix whose rows are linear combinations of the rows of $X$ and $XB$ means the same for the columns. At the same time, $|\cdot|F$ simply means the norm of the given matrix when viewed as a vector (by, say, writing the columns on top of one another). Thus, one could choose $C^{-1}$ to arbitrarily modify the columns, and then $C(XC^{-1})$ will modify the _rows of the transformed matrix, which does not seem intuitively to yield any meaningful result. Notice how this interpretation gives back the orthogonal matrix since that preserves norms. – William M. Aug 12 '21 at 14:51
  • you should confirm that you are working over $\mathbb R$. The result is true and it is an inequality due to Schur -- equality occurs iff $\big(CXC^{-1}\big)$ is normal. – user8675309 Aug 12 '21 at 16:49
  • @user8675309 Do you have a reference for this result? – Alphie Aug 12 '21 at 17:05
  • @Alphie this is the kind of thing typically left as an exercise in a lot of intro linear algebra texts e.g. Ken Kuttlers. Typically the sequence is, working over $\mathbb C$ prove normality $\iff$ unitary diagonalizability $\iff$ meeting said inequality of Schur with equality. A big hint for Schur's Inequality and the second "$\iff$": apply Schur Unitary Triangularization – user8675309 Aug 12 '21 at 17:13
  • @Alphie in my opinion this should have been left as an exercise for OP or marked as duplicate – user8675309 Aug 12 '21 at 17:42

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Regard $C$ and $X$ as complex $n\times n$ matrices. Since $CXC^{-1}$ and $X$ are similar, they have the same eigenvalues, which are the real diagonal entries of $X$.

Then use the Schur inequality from here to obtain the desired result. Equality obtains if and only if $CXC^{-1}$ is normal.

Alphie
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