# On one property of positive semidefinite matrices

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?

## 1 Answer

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 $x_i,x_j$.

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