0
$\begingroup$

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).

$\endgroup$
5
  • $\begingroup$ 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 $\mathbf E$? $\endgroup$ Commented May 12, 2012 at 5:16
  • $\begingroup$ @J.M. $E$ is a diagonal matrix is the only condition. $\endgroup$ Commented May 12, 2012 at 5:17
  • $\begingroup$ @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. $\endgroup$ Commented May 12, 2012 at 5:18
  • $\begingroup$ Related .. math.stackexchange.com/questions/4336/… $\endgroup$
    – Dilawar
    Commented May 12, 2012 at 5:44
  • $\begingroup$ So, you have answered your question, right? $\endgroup$ Commented May 12, 2012 at 6:32

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .