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

**References**:**Difference between scalar and vector inequality!***From:*"Sungjin Kim" <kimsj@mobile.snu.ac.kr>