Inequality

**usagi_killer** · March 21st 2011, 04:51 AM

Hi, I was wondering what values of a and b such that the entries for this matrix are all non negative?

What I did was:

we know that 4b-1<0

so we must have 4b=>0 and b+2a=>0 and 8a+1=>0

then we get 0<=b<1/4 and a=>-1/8 and a=>b/-2

we can find an interval for b but what about a?

cheers

**FernandoRevilla** · March 21st 2011, 08:20 AM

1. Independently of $a,b$ we have $a_{11},a_{12},a_{13}$ non negative.

2. $a_{22},a_{23},a_{33}$ are non negative iff $4b-1<0$ i.e. $b<1/4$ .

3. For $b<1/4$ , $a_{32}$ is non negative iff $b\geq 0$ .

4. For $0\leq b<1/4$ , $a_{21}$ is non negative iff $8a+1\geq 0$ i.e. $a\geq -1/8$ .

5. For $0\geq b<1/4$ and $a\geq -1/8$ , $a_{31}$ is non negative iff $b+2a\geq 0$ .

So, the solution is the region $R$ of the $ab$ plane:

$R \equiv\begin{Bmatrix}2a+b\geq 0\\a\geq -1/8\\0\leq b<1/4\end{matrix}$

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