Is This Norm Equality Correct for \( Ax \)?

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The discussion centers on the verification of the norm equality \( \|x\|_2 \|A\|_2 = \|Ax\|_2 \). It is established that this equality is incorrect when using the operator norm defined as \( \|A\| = \sup_{\|x\|=1} \|Ax\| \). A counterexample is provided with \( x = (1,0,...,0) \), demonstrating that the left-hand side encompasses components beyond those in the first column of matrix \( A \). The conclusion suggests that the correct relationship should involve an inequality, specifically \( \|x\|_2 \|A\|_2 \leq \|Ax\|_2 \).

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I just want to verify if the following is correct
\left\right\|x\|2.\left\right\|A\|2= \left\right\|Ax\|2

Thanks
 
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How are you defining the norm of the operator? As Tr(A^TA)? In that case, no. If you let x=(1,0,...,0), the right-hand side only contains components from the first column of A but the left-hand side contains other components.

However, the norm of A is often defined as

\|A\|=\sup_{\|x\|=1}\|Ax\|

so maybe if you change the = to ≤...
 

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