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Math Help - Positive definite matrices

  1. #1
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    Positive definite matrices

    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)?
    A=\begin{pmatrix}<br />
s & -4 & -4 \\ -4 & s & -4 \\ -4 & -4 & s<br />
\end{pmatrix} and B=\begin{pmatrix}<br />
t & 3 & 0 \\ 3 & t & 4 \\ 0 & 4 & t<br />
\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?
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  2. #2
    Senior Member roninpro's Avatar
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    There is another characterization of positive definiteness:

    x^T A x>0 for all column vectors x.

    This might be a little easier to use.
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  3. #3
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    Quote Originally Posted by roninpro View Post
    There is another characterization of positive definiteness:

    x^T A x>0 for all column vectors 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...
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