A simple "non" matrix norm

Jim Arvo · **Joined:** Fri, 10 Sep 2010 19:17:14 UTC **Posts:** 3

I recently came upon the need for an unusual norm on the vector space of matrices:

|| M || = max || m_i ||_p

where M is an mxn matrix (real or complex) and the "max" is over all the columns of the matrix, m_0, ... m_n. That is, this norm is the maximal vector p-norm among all the columns of M. This is *not* a true matrix norm, in the usual sense, as it is not submultiplicative, but it *is* a true vector norm.

I have several questions:

1) Has anyone come across this norm before? I've been unable to find anything through various searches.

2) Has it appeared in the literature? (If so, a reference would be much appreciated.)

3) Is there any common terminology for non-submultiplicative norms on matrices? ("pseudo-norm" or "semi-norm" would do nicely, but they are already used for other purposes.)

helmut · **Posted:** Fri, 10 Sep 2010 21:22:24 UTC

People usually ignore the matrix structure and view this as a direct product of the column vectors.

Notation is something like $\[ \left(\bigoplus \ell_n^p\right)^q\]$ for $1\leq p,q \leq\infty$ .

outermeasure · **Posted:** Sat, 11 Sep 2010 04:18:20 UTC

Jim Arvo wrote:

I recently came upon the need for an unusual norm on the vector space of matrices:

|| M || = max || m_i ||_p

where M is an mxn matrix (real or complex) and the "max" is over all the columns of the matrix, m_0, ... m_n. That is, this norm is the maximal vector p-norm among all the columns of M. This is *not* a true matrix norm, in the usual sense, as it is not submultiplicative, but it *is* a true vector norm.

I have several questions:

1) Has anyone come across this norm before? I've been unable to find anything through various searches.

2) Has it appeared in the literature? (If so, a reference would be much appreciated.)

3) Is there any common terminology for non-submultiplicative norms on matrices? ("pseudo-norm" or "semi-norm" would do nicely, but they are already used for other purposes.)

Well, your norm is the operator norm $\lVert M\rVert_{\ell_1\to\ell_p}$ .

helmut wrote:

People usually ignore the matrix structure and view this as a direct product of the column vectors.

Notation is something like $\[ \left(\bigoplus \ell_n^p\right)^q\]$ for $1\leq p,q \leq\infty$ .

Or $\ell_n^q(\ell_m^p)$ .

Jim Arvo · **Joined:** Fri, 10 Sep 2010 19:17:14 UTC **Posts:** 3

Hello helmut. Thanks for the quick response. Indeed, what I am doing can be expressed as the direct product of linear spaces with different p-norms (the ones I am most interested in are p=2 and p=infinity), which itself has a "natural" norm; namely, the direct product of the norms. This direct product arises naturally from a problem I am working on which, in turn, suggests the norm I asked about. I'm interested in finding out whether this particular type of norm has surfaced in other applications, and if so, whether its failing to meet the additional axiom usually required of a matrix norm (namely ||AB|| <= ||A|| ||B||) presents any difficulty.

I was also asking about terminology, but I just stumbled upon several references that provided me with the answer to that part: such norms are occasionally called "vector matrix norms", or more commonly "generalized matrix norms". Knowing what they are called is obviously a great help in finding relevant papers, but I have still not located any that address the norm I described.

Jim Arvo · **Joined:** Fri, 10 Sep 2010 19:17:14 UTC **Posts:** 3

Hi Outermeasure. If I'm understanding your notation correctly, you are saying that this norm is the finite-dimensional analog of the operator norm on the space of linear functions from $\ell_l$ to $\ell_p$ . (Actually, that should be $\ell_\infty$ to $\ell_p$ , right?) But all operator norms are automatically submultiplicative, just as all induced matrix norms are, right? If so, then the norm I defined is not an operator norm, and that's exactly my concern. Here's another way to see the difference. Let's call my norm ${\| \cdot \|}_p^{\ast}$ . Then

${\| A \|}_p^{\ast} = \max \frac{ {\| A e_i \|}_p }{ {\| e_i \|}_\infty} \le \sup \frac{ {\| A x \|}_p }{ {\| x \|}_\infty} = {\| A \|}_{\ell_\infty \rightarrow \ell_p}$

where e_i

denotes the ith column of the identity matrix. So my norm is clearly bounded above by the operator norm that you suggested, but the equality cannot hold in general because the latter is submultiplicative and the former is not (except when p=1). Am I missing or misinterpreting something?

outermeasure · **Posted:** Mon, 13 Sep 2010 05:52:10 UTC

Jim Arvo wrote:

Hi Outermeasure. If I'm understanding your notation correctly, you are saying that this norm is the finite-dimensional analog of the operator norm on the space of linear functions from $\ell_l$ to $\ell_p$ . (Actually, that should be $\ell_\infty$ to $\ell_p$ , right?) But all operator norms are automatically submultiplicative, just as all induced matrix norms are, right? If so, then the norm I defined is not an operator norm, and that's exactly my concern. Here's another way to see the difference. Let's call my norm ${\| \cdot \|}_p^{\ast}$ . Then

${\| A \|}_p^{\ast} = \max \frac{ {\| A e_i \|}_p }{ {\| e_i \|}_\infty} \le \sup \frac{ {\| A x \|}_p }{ {\| x \|}_\infty} = {\| A \|}_{\ell_\infty \rightarrow \ell_p}$

where e_i

denotes the ith column of the identity matrix. So my norm is clearly bounded above by the operator norm that you suggested, but the equality cannot hold in general because the latter is submultiplicative and the former is not (except when p=1). Am I missing or misinterpreting something?

No! It is the operator norm for $(\mathbb{R}^n,\lVert.\rVert_1)\to(\mathbb{R}^m,\lVert.\rVert_p)$ . I'll leave you to prove it is the case.

Also, it is not submulticative, and it makes no sense to multiply because they m and n are not equal (which is why you say M is an mxn matrix). Even if they are equal, you still have the problem of $p\neq 1$ . Of course, if p=1 and m=n, then you do indeed have the usual Banach algebra norm on $\mathop{\mathrm{End}}(\mathbb{R}^n,\lVert.\rVert_1)$ .

S.O.S. Mathematics CyberBoard

A simple "non" matrix norm

Who is online