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Could someone show me an example of a $3\times 3$ hermitian semidefinite positive matrix? (I mean not only symmetric, but hermitian, with complex coefficients)

There is a lot of work on hermitian semidefinite positive matrices, but it seems that people talk about them and don't take the time to exhibit examples.

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  • $\begingroup$ All such matrices are unitarily equivalent to diagonal matrices with real non-negative entries. $\endgroup$ – DisintegratingByParts Jan 11 '18 at 19:43
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Note that a Hermitian matrix will be positive definite if and only if it has positive eigenvalues. A good "recipe" to quickly build a positive definite matrix is to select diagonal elements that are sufficiently big, since selecting large positive diagonal entries will guarantee positive eigenvalues (positive, since the eigenvalues of a Hermitian matrix are necessarily real). An example of a more precise (and useful) description of this phenomenon is given by the Gershgorin circle theorem.

So with that being said: without computing eigenvalues explicitly, I can guarantee that the Hermitian matrix $$ \pmatrix{100 & 1 - 2i & 3 + 5i\\ 1 + 2i & 100 & -9 + i\\ 3 - 5i & -9 - i & 100} $$ is positive definite.

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  • $\begingroup$ Yes it is! Thank you for the recipe. I wouldn't have thought using the Gershgorin theorem, but now, it's obvious : if the $R_i$ are enough small and the $a_{ii}$ enough large, the corresponding disk is necessarily included in $\{x>0,y>0\}$. Bright! $\endgroup$ – Andrew Jan 11 '18 at 15:55
  • $\begingroup$ Oops, maybe I got too excited... Sorry, I have to investigate further the link with Gershgorin. $\endgroup$ – Andrew Jan 11 '18 at 16:06
  • $\begingroup$ @Andrew you seem to have the right idea. And again, because the matrix is Hermitian, the eigenvalues are necessarily real. $\endgroup$ – Omnomnomnom Jan 11 '18 at 17:56
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Hint: The identity matrix is Hermitian and psd.

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