Is ||x||₂ · ||A||₂ = ||Ax||₂?

[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 ≤...