Wikipedia tells me a little about it.

Following the wiki-link treasure hunt leads me to topics such as "p-norms on finite dimensional vector spaces".

Which makes me want to ask: what's a good textbook or reference where I can learn more about the p-norms on finite dimension vector spaces? I need to learn enough in order to answer the following questions given a proposed induced norm:

0) Why is it that in general $\|Ax\| \leq \|A\|\|x\|$ (where $\|A\|$ is the matrix norm, and $\|x\|$ is the vector norm)?

1) is the proposed induced norm a norm? (I think I know how to check for this now, as the Wikipedia page tells me what requirements a norm must satisfy to be considered one.)

2) given a proposed induced norm on a matrix, what is the corresponding vector norm?

Could you recommend a good resource to learn (notes, or textbook, or whatever)?

A textbook that is self-contained would be particularly valuable, but if I had to choose between an insightful text that was not self-contained, and a needlessly opaque text that was self-contained, I would pick the latter.


0) $\|Ax\|\leq \|A\|\|x\|$ because $\|A\|$ is essentially defined to be the smallest number $C$ for which $\|Ax\|\leq C\|x\|$ holds. For $x\neq0$ this directly follows from the usual definition and the definition of supremum. Just divide by $\|x\|$ and take the supremums on both sides.

1) The induced norm is always a matrix norm, the axioms follow from the corresponding properties for the vector norm. Induced norms have extra properties like $\|I\|=1$ and $\|AB\|\leq \|A\|\|B\|$, so the converse is not true.

2) Not every matrix norm is induced, so for some of them there is no corresponding vector norm. Vector norms that differ by a constant multiple induce the same matrix norm, as the definition implies by inspection. If you know in advance that a given norm is induced, the corresponding vector norm can be recovered up to a constant multiple by taking norms of rank $1$ matrices, i.e. the ones with the range spanned by a single vector.

Lancaster and Tismenetsky's Theory of Matrices has a chapter on matrix and vector norms that goes into the details of the relationship between them, and $p$-norms are used in examples.

  • $\begingroup$ Actually, item 2 does not work with all rank 1 matrices with the range equal to the span of the given vector. However, if, e.g., $X=[x,0,\ldots,0]$, one can easily show that $\|X\|=\max_{y\neq 0}\frac{\|Xy\|}{\|y\|}=\frac{\|Xe_1\|}{\|e_1\|}=\frac{\|x\|}{\|e_1\|}$. $\endgroup$ – Algebraic Pavel Sep 5 '14 at 12:59
  • $\begingroup$ @Algebraic Pavel Let $A_{u,v}$ denote the matrix that acts on vectors as $A_{u,v}x:=(u,x)v$, where $(u,x)$ is the dot product. Then by definition of induced norm $\|A_{u,v}\|=\sup_{x\neq0}\frac{|(u,x)|}{\|x\|}\|v\|=\|u\|^*\|v\|$, where $\|\cdot\|^*$ is the dual norm. Fixing any $u\neq0$ we recover the vector norm for all $v$ up to a multiple by taking $\|A_{u,v}\|$. $\endgroup$ – Conifold Sep 6 '14 at 19:57
  • $\begingroup$ True. Thanks for the clarification. $\endgroup$ – Algebraic Pavel Sep 7 '14 at 23:55

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