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A symmetric matrix is positive definite iff all eigenvalues are positive. I have been given a 3X3 symmetric matrix. I have calculated the eigenvalues two of which are negative. Does this mean this matrix is negative definite or indefinite (as it is certainly not positive definite)?

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    $\begingroup$ With one positive and two negative eigenvalues it is called indefinite. With one eigenvalue $0$ and two negative it is called negative semidefinite. $\endgroup$
    – Will Jagy
    Feb 4 '14 at 4:48
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If two are negative and the third is positive, then the matrix is sign indefinite.

If two are negative and the third is zero, then it is negative semi definite.

You can justify this by noting that $$ \lambda_{\text{min}} \left|| x \right|| \le x^t A x \le \lambda_{\text{max}} \left|| x \right||$$

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