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Re: Difference between scalar and vector inequality!
*To*: mathgroup at smc.vnet.net
*Subject*: [mg52283] Re: [mg52276] Difference between scalar and vector inequality!
*From*: DrBob <drbob at bigfoot.com>
*Date*: Sun, 21 Nov 2004 07:23:18 -0500 (EST)
*References*: <200411200841.DAA08798@smc.vnet.net> <opshrfvlt8iz9bcq@monster.cox-internet.com>
*Reply-to*: drbob at bigfoot.com
*Sender*: owner-wri-mathgroup at wolfram.com
I had an error in the definition of cc that made it equal to bb.
The counter-example is still valid, but here's another, without that restriction:
aa = Array[a, {2, 2}];
bb = Array[b, {2, 2}];
cc = Array[c, {2, 2}];
ii = IdentityMatrix[2];
counter = First@FindInstance[{Det[ii + aa + cc] < Det[
ii + bb + cc], Det[ii + aa] > Det[ii + bb] > 0, Det@aa > 0, Det@bb > 0, \
Det@cc > 0}, Flatten@{aa, bb, cc}];
aa /. counter
{{-2, 0}, {-1, -3}}
bb /. counter
{{-2, 0}, {-1, -2}}
cc /. counter
{{3/2, 2}, {-1, -1}}
All the determinants are positive, so the Logs are defined:
Det /@ {ii + aa, ii + bb, aa,
bb, cc, ii + aa + cc,
ii + bb + cc} /. counter
{2, 1, 6, 4, 1/2, 5/2, 3}
Bobby
On Sat, 20 Nov 2004 05:07:59 -0600, DrBob <drbob at bigfoot.com> wrote:
> No. Here's a counterexample:
>
> aa = Array[a, {2, 2}];
> bb = Array[b, {2, 2}];
> cc = Array[b, {2, 2}];
> ii = IdentityMatrix[2];
> counter = First@FindInstance[{Det[ii + aa + cc] < Det[
> ii + bb + cc], Det[ii + aa] > Det[ii + bb], Det@aa > 0, Det@bb > 0, \
> Det@cc > 0}, Flatten@{aa, bb, cc}]
> {Det[ii + aa + cc] - Det[ii + bb + cc],
> Det[ii + aa] - Det[ii + bb]} /. counter
>
> {a[1, 1] -> -(833/128),
> a[1, 2] -> -58, a[2, 1] -> -1,
> a[2, 2] -> -9, b[1, 1] -> -13,
> b[1, 2] -> -19, b[2, 1] -> -2,
> b[2, 2] -> -3}
> {-(53/128), 1/16}
>
> aa /. counter
> {{-(833/128), -58}, {-1, -9}}
>
> bb /. counter
> {{-13, -19}, {-2, -3}}
>
> cc /. counter
> {{-13, -19}, {-2, -3}}
>
> Bobby
>
> On Sat, 20 Nov 2004 03:41:53 -0500 (EST), Sungjin Kim <kimsj at mobile.snu.ac.kr> wrote:
>
>> The following inequality under given condition is true for scalar. However,
>> is this still true for matrix?
>>
>> log| I + A + C| >= log| I + B + C|
>> if log| I + A| >= log| I + B| and A, B, C >= 0
>> where |A| is absolute and determinant for scalar and matrix, respectively,
>> and A >= 0 means semi positive scalar or semi positive definite matrix,
>> respectively.
>>
>> Furthermore, is it possible to prove it using our Mathematica?
>> Thank you in advance.
>>
>> Br,
>> - Sungjin Kim
>> communication at samsung.com
>> kimsj at mobile.snu.ac.kr
>>
>>
>>
>>
>
>
>
--
DrBob at bigfoot.com
www.eclecticdreams.net
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