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

[ahamdiheme]

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

Thanks

 

[Fredrik]

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 ≤...