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 >