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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: [mg52284] Re: Difference between scalar and vector inequality!
  • From: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>
  • Date: Sun, 21 Nov 2004 07:23:19 -0500 (EST)
  • References: <cnn1ct$8t4$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Here is a counterexample.

Define the a, b, c matrices (2 by 2 is sufficient for a counterexample):

{a,b,c}={{{0.24298622797043015,-0.12198035733739936},{-0.12198035733739934,
        0.9450334658185751}},{{0.6585224329295042,-0.02046920647085003},{-0.\
020469206470850036,
        0.42623089412448895}},{{0.4810967067675747,-0.25291802786229},{-0.\
25291802786229,0.8548710119220764}}};

Evaluate the determinants and the two inequalities:

i=IdentityMatrix[2];
{Det[a],Det[b],Det[c],Log[Det[i+a]]>=Log[Det[i+b]],
  Log[Det[i+a+c]]>=Log[Det[i+b+c]]}

This produces the output:

{0.214751,0.280264,0.347308,True,False}

The |a|, |b| and |c| are all non-negative.
log |i+a| >= log |i+b| is true
log |i+a+c| >= log |i+b+c| is false

BTW you don't need the Log functions above because the logarithm function is 
monotonic and so doesn't affect the inequality test.

Steve Luttrell

"Sungjin Kim" <kimsj at mobile.snu.ac.kr> wrote in message 
news:cnn1ct$8t4$1 at smc.vnet.net...
> 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
> 



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