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Positive Semidefinite Rank


The real positive semidefinite rank of a nonnegative matrix A=(a_(ij)) is the smallest positive integer r admitting real symmetric positive semidefinite matrices X_i,Y_j of size r×r with

 a_(ij)=Tr(X_iY_j),

where Tr is the matrix trace (Fawzi et al. 2015). A matrix whose entries are all the same positive number has positive semidefinite rank 1.


See also

Matrix Rank, Nonnegative Matrix, Positive Semidefinite Matrix, Semidefinite Lift, Semidefinite Programming

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References

Fawzi, H.; Gouveia, J.; Parrilo, P. A.; Robinson, R. Z.; and Thomas, R. R. "Positive Semidefinite Rank." Math. Program. 153, 133-177, 2015. https://doi.org/10.1007/s10107-015-0922-1.

Cite this as:

Weisstein, Eric W. "Positive Semidefinite Rank." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PositiveSemidefiniteRank.html

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