# Positive definite matrices

#### TheFangel

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?

#### roninpro

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.

• TheFangel

#### TheFangel

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...