Re: Matrix Norm

On Jun 27, 6:37 am, buan <emailpuny...@xxxxxxxxxxx> wrote:
I get confused of this definition:
||A|| = sup ||Ax||/||x||
can anyone explain about it?
what is it mean? and also where that come from?
Henry

====

In vector space, the norm ||x|| of a vector x describes the "size" or
"length" of the vector.

An operator maps vectors from one vector space to another.
Intuitively, if an operator A tends to map to "small" vectors (vectors
with small norms), then we might say that the size of operator A is
"small". If operator A tends to map to large vectors, then we might
say that the size of vector A is "large".

This is at least partly what operator norms are all about. If A is an
operator that maps from vector space X to vector space Y, then the
norm of A is
|||A||| = sup{ ||Ax|| : x in X and ||x||<=1 }

In words, if we restrict ourselves to vectors x with small norms (||x||
<=1), then the largest norm ||Ax|| that A can produce is the norm of
A.

(Notice above that I used the triple bar notation |||.||| for the
operator norm --- this is due to Horn and Johnson, "Matrix Analysis")

There are some interesting results related to this:
1. |||.||| is a legitimate norm --- that is,
|||A||| >= 0 (non-negative)
|||A||| = 0 iff A=0 (non-degenerate)
|||aA||| = |a| |||A||| (homogeneous)
|||A+B||| <= |||A||| + |||B||| (subadditive)

2. The set of all linear operators with |||.||| form a normed vector
space (normed vector space of linear operators) --- that is,
A+B is an operator
aA is an operator
A+B = B+A
(A+B)+C = A+(B+C)
...

3. |||I||| = 1
4. |||Ax||| <= |||A||| ||x||
5. |||AB||| <= |||A||| |||B|||

Here is a document with some proofs and references related to operator
norms --- you can download it if you have interest:
http://banyan.cm.nctu.edu.tw/~dgreenhoe/msd/opnorm.pdf

Dan Greenhoe

.

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