How do we know that the Frobenius norm is not subordinate to a vector norm?

252 Views Asked by At

In a quiz

The following statement was said to be False: "The Frobenius norm is a matrix norm subordinate to (or induced by) a vector norm"

At the time of writing, I can't see why/how we know this to be False. How do we know that there isn't some vector norm to which the Frobenius norm is subordinate?

the operational definition of subordinate norm being:

"Given a vector norm $||\cdot||$, the corresponding subordinate matrix norm is defined by $$||A||= \max_{x \not =0} \frac{||Ax||}{||x||}$$"