an:06945438
Zbl 1408.15016
Fawzi, Hamza; Gouveia, Jo??o; Robinson, Richard Z.
Rational and real positive semidefinite rank can be different
EN
Oper. Res. Lett. 44, No. 1, 59-60 (2016).
00414783
2016
j
15B48
matrix factorization; positive semidefinite rank; semidefinite programming
Summary: Given a $p \times q$ nonnegative matrix $M$, the psd rank of $M$ is the smallest integer $k$ such that there exist $k \times k$ real symmetric positive semidefinite matrices $A_1, \ldots, A_p$ and $B_1, \ldots, B_q$ such that $M_{i j} = \langle A_i, B_j \rangle$ for $i = 1, \ldots, p$ and $j = 1, \ldots, q$. When the entries of $M$ are rational it is natural to consider the rational-restricted psd rank of $M$, where the factors $A_i$ and $B_j$ are required to have rational entries. It is clear that the rational-restricted psd rank is always an upper bound to the usual psd rank. We show that this inequality may be strict by exhibiting a matrix with psd rank four whose rational-restricted psd rank is strictly greater than four.