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 July 1st, 2013, 04:21 AM #1 Newbie   Joined: Jul 2013 Posts: 2 Thanks: 0 S-lemma Hi everyone, I've joined this forum because of a exercise that is bugging me. I study control-theory, and the following question is in one of the exercises: $Let T_0 and T_1 be two symmetric matrices and consider the following two conditions i) \begin{equation} x^TT_0x > 0 \end{equation} whenever \begin{equation} x^TT_1x > 0 \end{equation} for x\neq0 and >0 means positive definite. ii) there is a \beta > 0 such that: \begin{equation} T_0 ? \betaT_1 > 0 \end{equation} It can fairly easy be proven that the second condition implies the first. If however there exists a x_0 for which x_0 T_1 x_0 > 0 then also the first condition implies the second. Can you give an example to explain the need of the extra condition (slater's condition)?$ I've tried to reason it through, to find some situation in which the extra condition is needed, but for every situation it seems the condition will hold anyway... Can somebody help me with this? I am not asking for the right answer, but just for some indication in which direction I should look. Thank you in advance!
July 1st, 2013, 04:27 AM   #2
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Re: S-lemma

Sorry for the error in the previous post. Here is the lemma:

[attachment=0:69ff37x8]snip.PNG[/attachment:69ff37x8]

The question is to find an example which explains the need of the extra condition on $T_1$ for the first to imply the second.
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