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Let $A,B,C$ complex $n\times n$ Matrices with ones on the diagonal and entries of absolute value 1. Further, let $$A=(a_{ij})_{i,j=1}^n,\; B=(b_{ij})_{i,j=1}^n \mbox{ and } C=(a_{ij}b_{ij})_{i,j=1}^n.$$

Can we say something about the positive semi-definiteness of $A$ if we know that $B$ and $C$ have this property?

Any hint to a similar problem or result or counter-example would be greatly appreciated.

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    $\begingroup$ Are $A$, $B$, and $C$ related somehow? Why should positive semi-definiteness of $A$ depend at all on $B$ and $C$? $\endgroup$ – User8128 Jun 28 '17 at 17:43
  • $\begingroup$ @User8128 Thank you. I forgot something ... edited the question $\endgroup$ – Vincent.W. Jun 28 '17 at 17:46
  • $\begingroup$ Got it. Thanks makes more sense! $\endgroup$ – User8128 Jun 28 '17 at 17:46
  • $\begingroup$ Not sure about the star notation. Is $C$ equal to the Hadamard product of $A$ and $B$? $\endgroup$ – Paul Aljabar Jun 28 '17 at 17:51
  • $\begingroup$ @PaulAljabar Yes.. its just a componentwise product. $\endgroup$ – Vincent.W. Jun 28 '17 at 17:53
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If by complex matrices you mean Hermitian matrices the answer is yes, for a silly reason and a non-silly reason. Suppose $B$ and $C$ are PSD with $1$s on the diagonal and all off diagonal entries of absolute value $1$. You ask if the elementwise quotient $A=(c_{i,j}/b_{i,j})$ is PSD. The silly reason is that the complex conjugate of a complex number of absolute value $1$ is its reciprocal. So the matrix $D=(\bar{b}_{i,j})$ is the matrix of reciprocals of the entries in $B$, and $A$ is the element-wise product of the two PSD matrices $A$ and $D$. Which it is also PSD, by the non-silly Schur product theorem: https://en.wikipedia.org/wiki/Schur_product_theorem.

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  • $\begingroup$ Thank you! However, by complex matrix I mean a matrix with complex entries-no further regularity assumed. $\endgroup$ – Vincent.W. Jun 29 '17 at 6:54
  • $\begingroup$ I mean, what is a complex positive definite matrix for you? Do you require $\bar z' M z\ge 0$ for all complex vectors $z$, with $\bar z'$ denoting the conjugate transpose? If so, the argument I supplied answers your question in the affirmative. $\endgroup$ – kimchi lover Jun 29 '17 at 15:02
  • $\begingroup$ Yes, its just that you wrote that the answer is yes for hermitian matrices. But,if $B$ is psd and not hermitian then also its transpose conjugate $D$ will be psd. However then you dont have $B=D$ and hence $C\circ D\neq A$ ($\circ$ denotes hadamard) since $c_{i,j}*b^{-1}_{j,i}\neq a_{i,j}$. $\endgroup$ – Vincent.W. Jun 29 '17 at 15:44
  • $\begingroup$ Can you tell me precisely what you mean by PSD, and supply an example of a non-Hermitian PSD matrix? The [Wikipedia article]( en.wikipedia.org/wiki/Positive-definite_matrix) does not help me understand more clearly what you are thinking about. $\endgroup$ – kimchi lover Jun 29 '17 at 17:58
  • $\begingroup$ You are right. Psd implies hermitian. $\endgroup$ – Vincent.W. Jun 29 '17 at 18:41

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