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MathGroup Archive 2004

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Re: Difference between scalar and vector inequality!

  • To: mathgroup at smc.vnet.net
  • Subject: [mg52282] Re: [mg52276] Difference between scalar and vector inequality!
  • From: DrBob <drbob at bigfoot.com>
  • Date: Sun, 21 Nov 2004 07:23:17 -0500 (EST)
  • References: <200411200841.DAA08798@smc.vnet.net>
  • Reply-to: drbob at bigfoot.com
  • Sender: owner-wri-mathgroup at wolfram.com

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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