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Suppose that $A = \begin{bmatrix} 1 & 4 & 3 \\ 4 & 2 & 5 \\ 3 & 5 & 3 \end{bmatrix}$

Also suppose that I add a diagonal matrix $E$ to $A$ (that is consider $A+E$). If all the eigenvalues of $A+E$ are positive, will it be positive definite?

Edit. Adding a symmetric matrix to a diagonal matrix will be a symmetric matrix. So I can just add a large diagonal matrix get a positive definite matrix (e.g. so that all the eigenvalues are positive).

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    So, you have answered your question, right?2012-05-12

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A symmetric matrix is positive definite if and only if all of its eigenvalues are positive. Therefore: if you add a diagonal (or just a symmetric) matrix $E$ to $A$, and find that all the eigenvalues of $A+E$ are positive, the conclusion will be that $A+E$ is positive definite.