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!
$\endgroup$ 51 Answer
$\begingroup$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.
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