# Euclidean Norm

1. ### ahamdiheme

26
I just want to verify if the following is correct
$$\left\right\|x\|$$2.$$\left\right\|A\|$$2= $$\left\right\|Ax\|$$2

Thanks

2. ### Fredrik

10,263
Staff Emeritus
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 ≤...