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Let $A \in \mathbb{C}^{n\times n}$, is a hermitian matrix with guaranteed real and positive diagonal entries. Also $\det(A)>0$. Is $A$ positive-definite?

Positive-definite matrices have real and positive diagonal entries, but I have not found any resource whether the converse is true or not. I know that this is true in case of eigenvalues, but I wish not to calculate eigenvalues. This question solves it for a real matrix and I was wondering if the same can be done for complex matrix?

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    $\begingroup$ in that case $A=I + B B^H$ is positive definite, because $B B^H$ is positive semidefinite, and $I$ is positive definite. $\endgroup$
    – orangeskid
    Jan 10, 2020 at 15:15
  • $\begingroup$ @orangeskid thanks so much. I should have properly explained in my question. $\endgroup$
    – impopularGuy
    Jan 10, 2020 at 15:42

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The linked question only solves it for diagonally dominant matrices, and the same argument made there could be made for diagonally dominant complex matrices.

In general, the answer to your question is no, even for real matrices. For example, the matrix $$\begin{bmatrix}1&-3 &&-3\\ -3 &1&&2\\-3&&2&&1\end{bmatrix}$$ has two negative eigenvalues and one positive eigenvalue.

Note, of course, that the matrix I provide is not diagonally dominant.

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  • $\begingroup$ Thanks, I now understand. What else can be done for checking whether $A$ is positive-definite or not? $\endgroup$
    – impopularGuy
    Jan 10, 2020 at 14:19
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    $\begingroup$ @titusarmah99 The easiest way to show a matrix is positive definite is to calculate its Cholesky decomposition. en.wikipedia.org/wiki/Cholesky_decomposition $\endgroup$
    – 5xum
    Jan 10, 2020 at 14:20
  • $\begingroup$ In my case, $A$ is calculated as $A=BB^H$ where $B\in \mathbb{C}^{n\times m}$. How will it make sense to find Cholesky decomposition? $\endgroup$
    – impopularGuy
    Jan 10, 2020 at 14:34
  • $\begingroup$ Sorry, it is $A=I_n + BB^H$ where $I_n$ is identity matrix. $\endgroup$
    – impopularGuy
    Jan 10, 2020 at 14:42
  • $\begingroup$ @titusarmah99 A sum of positive definite matrices is positive definite. $\endgroup$
    – 5xum
    Jan 10, 2020 at 18:25

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