Positive definite matrices

May 2010
15
0
Hi,

I have this problem that I really can't solve.

For which s and t do A and B have all eigenvalues λ > 0 (therefore positive definite)?
\(\displaystyle A=\begin{pmatrix}
s & -4 & -4 \\ -4 & s & -4 \\ -4 & -4 & s
\end{pmatrix}\) and \(\displaystyle B=\begin{pmatrix}
t & 3 & 0 \\ 3 & t & 4 \\ 0 & 4 & t
\end{pmatrix}\)

I tried to use the characteristic polynomial but I can't solve it since it's third degree. I also tried Cholesky decomposition for symmetric matrices, but the elements of the matrices are too complicated to be used in a useful way.

Could someone help me?
 
Nov 2009
485
184
There is another characterization of positive definiteness:

\(\displaystyle x^T A x>0\) for all column vectors \(\displaystyle x\).

This might be a little easier to use.
 
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May 2010
15
0
There is another characterization of positive definiteness:

\(\displaystyle x^T A x>0\) for all column vectors \(\displaystyle x\).

This might be a little easier to use.
Thank you very much!
I'll try that out. Looks much easier than what I was using...