Definition - An entry wise non-negative matrix is said to be ID if $A^{or} = [a_{ij}^r]$ is PSD (positive semi definite) for all $r \geq 0$ i.e r being a non-negative real no.
I need to prove or disprove that :
If A and B are ID matrices \ a ) then A+B is an ID matrix
b) and if AB = BA(so that AB becomes hermitian) then AB is also ID.

I have some example of ID matrices which are min(i,j), gcd(i,j), any 2-square PSD matrix etc. After taking these examples it seems the result is positive. Any help? Thanks in advance.

  • $\begingroup$ Just to make it clear to me: Your $A^{or}$ is the matrix whose $ij$ entry is the $r$th power of the $ij$ entry of $A$? Or is matrix multiplication somehow involved in the definition of ID? $\endgroup$ – kimchi lover Nov 20 '17 at 12:32
  • $\begingroup$ yes, it's the $r$'th power of $ij$'th entry like you said. $\endgroup$ – veer singh panwar Nov 20 '17 at 14:59

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