0
$\begingroup$

For a matrix $A$, is there any relation between its operator norm and $\infty$-norm (defined as the maximum of the absolute value of all the entries)?

$\endgroup$
1
  • $\begingroup$ In the same linked wiki page there are some relations here. $\endgroup$
    – awllower
    Commented Jul 16, 2016 at 7:25

1 Answer 1

2
$\begingroup$

The fact:

Any norms in a given finite-dimensional vector space are equivalent.

And it is easy to check the space of $n\times m$ matrices is a finite-dimensional vector space; also the operator norm and infinity norm are really norms.

So they are equivalent!
i.e.

$$||A||\le C_1||A||_{\infty}$$ and $$||A||_{\infty}\le C_2||A||$$

and that might be the relation you want.

$\endgroup$
2
  • 6
    $\begingroup$ So are there explicit relations? Like inequalities between the two norms. I think this might better help the OP. $\endgroup$
    – awllower
    Commented Jul 15, 2016 at 10:31
  • $\begingroup$ math.stackexchange.com/questions/3722793/… $\endgroup$
    – ABIM
    Commented Jun 2 at 23:57

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .