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?
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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. |
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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. |
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