1
$\begingroup$

I am in trouble with the definitions of positive definite and positive semidefinite matrices. By definition, does the following implication hold?

$$\mbox{positive definite} \implies \mbox{positive semidefinite}$$

I guess yes. Look at the following example. Let

$$A = \begin{vmatrix} 4 &-2& 0\\ -2& 4& -2\\ 0&-2&4 \end{vmatrix}$$

It is a diagonally dominant matrix $(|a_{ii}|\geq \sum_{j\neq i}|a_{ij}|$ for all $i$), so it is semidefinite positive. But, if I compute the eigenvalues, they are all strictly positive which implies actually that it is definite positive.

My question is: everytime I find that a matrix is semidefinite positive, thus I have to use another criterion in order to try to understand if it is "less" that semidefinite positive (i.e. definite positive)? Or am I wrong with something?

Thank you in advance!

$\endgroup$
5
  • $\begingroup$ Yes, every positive definite matrix is also positive semidefinite. $\endgroup$
    – angryavian
    Dec 4, 2020 at 15:38
  • 1
    $\begingroup$ like every positive number is non-negative $\endgroup$ Dec 4, 2020 at 15:39
  • $\begingroup$ Thank you. So, everytime I find that a matrix is semidefinite positive, thus I have to use another criterion in order to try to understand if it is "less" that semidefinite positive (i.e. definite positive)? $\endgroup$
    – C. Bishop
    Dec 4, 2020 at 15:48
  • 1
    $\begingroup$ I would say it's "more" than semidefinite positive (the set of semidefinite positive matrices is strictly contained in the set of definite positive matrices) $\endgroup$ Dec 4, 2020 at 15:59
  • 4
    $\begingroup$ @J.W.Tanner If positive definite implies positive semidefinite, I think that it should be the contrary. $\endgroup$
    – C. Bishop
    Dec 4, 2020 at 16:19

1 Answer 1

3
$\begingroup$

Positive definite means "All eigenvalues are greater than zero."

Positive semi-definite means "All eigenvalues are greater than or equal to zero."

The first of these implies the second.

The second does not imply the first, as the all-zero matrix shows.

$\endgroup$
2
  • $\begingroup$ Clear, thank you. So, everytime I find that a matrix is semidefinite positive, thus I have to use another criterion in order to try to understand if it is "less" that semidefinite positive (i.e. definite positive)? $\endgroup$
    – C. Bishop
    Dec 4, 2020 at 15:48
  • $\begingroup$ Actually, it's more than semi-definite -- it's all the way to definite. But yes, if you know only that a matrix is positive semi-definite, you need to do MORE to show that it's positive definite. (AFter all, sometimes it's NOT actually positive definite, e.g., my example of the zero-matrix. $\endgroup$ Dec 4, 2020 at 20:05

You must log in to answer this question.

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