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Topic: Inequalities in matrices
Replies: 8   Last Post: Apr 23, 2013 5:04 PM

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quasi

Posts: 10,188
Registered: 7/15/05
Re: Inequalities in matrices
Posted: Apr 23, 2013 3:57 PM
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dullrich wrote:
>quasi wrote:
>>dullrich wrote:
>>>quasi wrote:
>>>>gyansorova wrote:
>>>>>
>>>>>If I have
>>>>>
>>>>>A>B where A and B are the same type of matrices -
>>>>>what does this mean?

>>>>
>>>>It usually means A[i,j] > B[i,j] for all i,j.

>>>
>>>Reasonable guess. But you shouldn't post guesses as though
>>>they were facts.

>>
>>Well, there are quite a few books on nonnegative matrices,
>>and all of them use the notation
>>
>> A >= 0
>>
>>to mean
>>
>> A[i,j] >= 0 for all i,j,
>>
>>so that was from memory, not a guess.

>
>Really? That seems very curious. What's an example of such
>a book?


Bapat & Raghavan
Nonnegative Matrices and Applications
CUP, 1997

Berman & Plemmons
Nonnegative Matrices in the Mathematical Sciences
AP, 1979

Minc
Nonnegative Matrices
Wiley, 1988

>>But now that I've checked one of them, it's clear that those
>>books don't extend the notation A >= 0 to the notation A >= B,
>>so for that, I stand corrected.
>>

>>>What it really means is that A - B is positive definite.
>>>Which means that
>>>
>>> <(A-B) x, x> > 0
>>>
>>>for all x <> 0.

>>
>>Ok.


quasi



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