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I have a matrix $M$ that is a sum of rank 1 positive semidefinite and negative semidefinite matrices. I would like to know whether it is positive semidefinite and, if not, what properties would be needed to make it positive semidefinite.

Let $m,n,r\in \mathbb{N}$ such that $m> n> r>0$, and consider a set $\{b_1,\ldots,b_{nr}\}$ of $nr$ $m$-dimensional distinct binary vectors. The $m\times m$ matrix $M$ is the following:

$$M=\underset{j=1}{\overset{r}{\sum}}\underset{i=1}{\overset{n}{\sum}}(j-2)(b_{ij}b_{ij}^{\top})$$

Additionally, I know that the entries of $M$ are all greater than 0.

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  • $\begingroup$ Perhaps you should look at the singular value decomposition (SVD). $\endgroup$ – copper.hat Jun 13 '18 at 20:17
  • $\begingroup$ But how could I use SVD? I don't know the values of the entries of $M$ and I don't see how to exploit the vectors $b_1,\ldots,b_{nr}$. Thanks. $\endgroup$ – ziotomd Jun 14 '18 at 12:33

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