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This question reveals some important properties of positive semidefinite matrices. It would be very useful in solving the SDP problem (where I encountered it). The question is quite straightforward.

Let $A$ be an $n \times n$ positive semidefinite matrix with positive diagonal elements. Is it possible that any of its $2 \times 2$ submatrices defined as:

$$ \begin{bmatrix} a_{ii} & a_{ij} \\a_{ji} & a_{jj}\end{bmatrix} $$

is not positive semidefinite?

And if every this $2\times2$ submatrix is positive semidefinite, is A necessarily positive semidefinite?

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The submatrix you mention is necessarily positive semidefinite. In particular, if $e_1,\dots,e_n$ are the standard basis vectors, then it suffices to note that $$ (x_ie_i + x_je_j)^TA(x_ie_i + x_je_j) = \pmatrix{x_i&x_j} \pmatrix{a_{ii} & a_{ij}\\ a_{ji} & a_{jj}}\pmatrix{x_i\\x_j}\geq 0 $$ for any choice of $x_i,x_j$.

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  • $\begingroup$ Thank you very much! Very good answer! $\endgroup$ – Ben Wu Sep 20 '16 at 22:46
  • $\begingroup$ What about the inverse? If every $ 2\times 2$ submatrix is positive semidefinite, is A necessarily a positive semidefinite matrix? $\endgroup$ – Ben Wu Sep 21 '16 at 1:35
  • $\begingroup$ I assume you're talking specifically about principal submatrices, then $\endgroup$ – Omnomnomnom Sep 21 '16 at 2:00
  • $\begingroup$ I suspect not, but it's a reasonable question to ask. I would start a new post for the inverse question. $\endgroup$ – Omnomnomnom Sep 21 '16 at 2:03
  • $\begingroup$ I'm quite certain it is not. It will conflict some result I already got from the data. $\endgroup$ – Ben Wu Sep 21 '16 at 2:09

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