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I have a square symmetric matrix. I want to check if it is positive definite. I understand that according to wikipedia if all the eigenvalues are positive, the matrix is positive definite. I don't know if this means $\lambda$>0 or if it means $\lambda \geq$0. Because I have $\lambda$=0.

According to wiki zero isn't positive or negative. I am just a little confused on this topic.

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2 Answers 2

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A Positive Definite has full rank: all its eigenvalues are strictly positive.

A square symmetric matrix with non-negative eigenvalues (i.e., eigenvalues that are positive or zero) is called Positive Semi-Definite (PSD).

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  • $\begingroup$ Thanks! That was very helpful! $\endgroup$
    – erncyp
    Mar 13, 2015 at 10:24
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A positive definite symmetric matrix has strictly positive eigenvalues. If $0$ were an eigenvalue, the matrix would be singular since its kernel would be non-zero, which contradicts positive definiteness.

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  • $\begingroup$ Thanks! That was very helpful! $\endgroup$
    – erncyp
    Mar 13, 2015 at 10:24

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