Problem on norm of a matrix.

Asked Jun 28 '19 at 13:52

Active Jun 28 '19 at 14:37

Viewed 107 times

Actually, we know that the matrix group $M (n,\Bbb R)$ can be viewed as a topological space with respect to a point of the respective co-ordinates in $\Bbb R^{n^2}$ and the norm function can be defined as $||A||$ corresponding to the norm of the co-ordinate of the matrix $A$ .And, under this norm, obviously $||A||=||A^t|| , A^t$ is transpose of $A$ .

Now, under the norm, $||A||=\sup_{\Vert x\Vert=1}\Vert Ax\Vert$, for any matrix $A$ in $M (n,\Bbb R)$, how to prove $||A||=||A^t||$ ?

[Obviously, the above norm is well-defined as $S^{n-1}$ is compact in $\Bbb R^n$ and $x→\Vert Ax\Vert$ is continuous.]

edited Jun 28 '19 at 14:37

Adam Chalumeau

3,333

asked Jun 28 '19 at 13:52

Rabi Kumar Chakraborty

2,305

Can please someone edit the text using Latex or MathJax as I am completely unable of doing it . – Rabi Kumar Chakraborty Jun 28 '19 at 14:04
1

The definition of the operator norm depends on the norm you pick for $\mathbb{R}^n$. Are you using the Euclidean norm? – cmk Jun 28 '19 at 14:40
To cmk, no Sir, it is not the Euclidean norm, rather it is the norm defined by the continuous function stated above . – Rabi Kumar Chakraborty Jun 28 '19 at 15:41
I meant, what is the norm for $|x|?$ If you take the operator norm induced by, say, the $\ell_1$ norm, then this is obviously not true. This question has already been marked as a duplicate, but still. – cmk Jun 28 '19 at 16:26
@cmk, sorry Sir, for my misunderstanding, yes , I do follow Euclidean norm for the operator, but Sir can you please answer about why the result doesn't hold for the L1 norm ??? – Rabi Kumar Chakraborty Jun 28 '19 at 19:08
1

If $|A|p$ is the operator norm induced by the $p$ norm, then $|A|_1=|A^T|\infty,$ and this does not have to be equal to $|A^T|_1$ (recall that the $1$ norm here is the max absolute column sum, so it's not hard to find a counterexample). – cmk Jun 28 '19 at 19:53
Let us continue this discussion in chat. – Rabi Kumar Chakraborty Jun 29 '19 at 08:36

Problem on norm of a matrix.

0 Answers0