A symmetric basis exchange replaces two matroid bases 
and 
by
 |
where 
, 
,
and both resulting sets are matroid
bases. Each element therefore occurs the same number
of times in the pair before and after the exchange.
The symmetric basis exchange theorem states that, for every pair of matroid bases and for all 
, at least one suitable 
exists (Brualdi 1969, pp. 165-166). This local existence
statement does not imply that repeated exchanges connect all pairs with the same
multiset union, as proposed in White's
conjecture.
