# Is every positive definite matrix also positive semidefinite?

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?

• Yes, every positive definite matrix is also positive semidefinite. Dec 4, 2020 at 15:38
• like every positive number is non-negative Dec 4, 2020 at 15:39
• 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)? Dec 4, 2020 at 15:48
• 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) Dec 4, 2020 at 15:59
• @J.W.Tanner If positive definite implies positive semidefinite, I think that it should be the contrary. Dec 4, 2020 at 16:19