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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 $\mathbf E$? – J. M. May 12 '12 at 5:16
@J.M. $E$ is a diagonal matrix is the only condition. – thomas james May 12 '12 at 5:17
@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. – thomas james May 12 '12 at 5:18
Related ..… – Dilawar May 12 '12 at 5:44
So, you have answered your question, right? – Gerry Myerson May 12 '12 at 6:32

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.

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