As a classic example of Khachiyan shows, some semidefinite programs (SDPs) have solutions whose size -- the number of bits necessary to describe them -- is exponential in the size of the input. Exponential size solutions are the main obstacle to solve a long standing open problem: can we decide feasibility of SDPs in polynomial time?

The consensus seems that large solutions in SDPS are rare. Here we prove that they are actually quite common: a linear change of variables transforms every strictly feasible SDP into a Khachiyan type SDP, in which the leading variables are large. As to ""how large"", that depends on the singularity degree of a dual problem. Further, we present some SDPs in which large solutions appear naturally, without any change of variables. We also partially answer the question: how do we represent such large solutions in polynomial space?