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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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    Depends on the $\mathbf E$ you're adding. The most negative eigenvalue of $\mathbf A$ is $\approx -3.04$, so for instance adding a matrix $c\mathbf I$, where $0 < c < 3.04$ would not yield a positive definite matrix. Nothing special about your $\mathbf E$?2012-05-12
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    @J.M. $E$ is a diagonal matrix is the only condition.2012-05-12
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    @J.M. Also adding a symmetric matrix to a diagonal matrix will be a symmetric matrix. And then if its eigenvalues are all positive then it will be positive definite.2012-05-12
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    Related .. http://math.stackexchange.com/questions/4336/if-eigenvalues-are-positive-is-the-matrix-positive-definite2012-05-12
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    So, you have answered your question, right?2012-05-12

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