Suppose A is a Hermitian invertible matrix with positive diagonal entries. When A will become a positive definite or its all the eigen values will be positive?
Hermitian and positive definite matrices
1
$\begingroup$
matrices
-
3Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – 2012-11-02
2 Answers
1
The statement is not true: take $A:=\pmatrix{1&a\\a&1}$, where $a>1$: the eigenvalues are $1+a$ and $1-a<0$, and $A$ is invertible.
However, what is true is that a positive definite (hermitian) matrix has positive eigenvalues.
-
0Thanks for this counter example. – 2012-11-02
1
If $A$ is strictly diagonally dominant, $i.e.$ if $|a_{ii}| > \sum_{i \neq j} |a_{ij}|$ for all $i$, and all of the diagonal entries of $A$ are positive then $A$ will be positive definite.
However, there can be matrices which are positive definite and are not strictly diagonally dominant. In general, $A$ will be positive definite iff all of its eigenvalues are positive.