I've been trying to understand matrix norms (full disclosure: school assignment, not looking for answers, just clarity!), and how they follow from vector norms - been awhile since I did much linear algebra, so i'm struggling a bit with the notation, in particular I'm solving in the general case that for matrix A (and any nonzero vector x)

$$\frac{||Ax||_1}{||x||_1} \le C$$

For C = maximum column sum of A

The part I don't think I understand is what ${||Ax||_1}$ actually means, relative to matrix A. Could someone help me to understand a bit better the notation, and to apply the matrix norm?

  • $\begingroup$ Note that $Ax$ is a vector, to be evaluated wrt I norm (sup?). If the question is about matrix norms though, you are likely asked to solve the inequality for the maximum non-zero vector $x$. $\endgroup$ – gnometorule Jan 30 '13 at 2:57
  • $\begingroup$ So $Ax$ would be, in a general case that max non-zero vector? or one of the vectors in A? I just don't really understand how $Ax$ and $x$ are related, I guess. $\endgroup$ – Stephen Young Jan 30 '13 at 3:07
  • $\begingroup$ What I meant is that the usual definition of a matrix norm is $\sup_{x \neq 0} \frac{||Ax||}{||x||} = \sup_{x: ||x|| = 1} ||Ax||$. It's the same $x$ in numerator and denominator, as you take the $\sup$ of the fraction. So write it down, forgetting about the $\sup$, for some such $x$ (just calculate the vector $Ax$); do some thinking for this fixed $x$; then take the supremum over all such $x$. It's probably easier to use the 2nd, equivalent definition ($||x|| = 1$). $\endgroup$ – gnometorule Jan 30 '13 at 3:20

Since it looks like you are interested in the matrix norm which is induced by a particular vector norm (take a look here) , there is a nice geometric interpretation for $\|A\|$:

$$\|A\|:=\max_{\|x\|=1}\|Ax\|,$$ that is, if you start with any unit vector, compute its image under the transformation $A$, and compute the norm of that image (which is a vector, so its computed via a vector norm), then $\|A\|$ is the largest such resulting value. In other words, $\|A\|$ is the maximum "stretch" that $A$ "does" to a unit vector.

As an example, suppose $A=\begin{bmatrix}1 & 2\\0 & 3\end{bmatrix}$, so $A:\mathbb{R^2}\to\mathbb{R}^2$, and we will consider $\mathbb{R}^2$ with the 2-norm. Then the matrix norm induced by the (vector) 2-norm described above is summarized graphically with this figure:

Mathematica graphics

Note the unit vectors on the left and then some representative images under $A$. The length of the longest such image is $\|A\|$ (induced by this vector norm).

  • 1
    $\begingroup$ awesome - this is a much better description than from my class notes, thanks! $\endgroup$ – Stephen Young Jan 30 '13 at 3:32
  • $\begingroup$ So the natural matrix norm is the largest vector ||Ax|| out of all possible unit vectors x? $\endgroup$ – Jaigus Mar 15 at 23:43

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