Chelsea s German defender Antonio Rudiger (AFP)
Madrid brought in Rudiger from Chelsea on a four-year contract. The Germany international will be officially introduced on Monday.
Madrid praised Rudiger's ``very commanding physical presence'' and his experience playing in England, Germany and Italy.
The 29-year-old central defender helped Chelsea in its run to titles in the Champions League, Europa League, Club World Cup, UEFA Super Cup and FA Cup.
He scored five goals in 54 matches with Chelsea last season, both a record for him with the club.
Rudiger made his Bundesliga debut with Stuttgart as an 18-year-old before joining Roma in 2015 and eventually Chelsea in 2017.
Real Madrid played the Champions League final against Liverpool with central defenders Eder Militao and David Alaba, who was signed a year ago from Bayern Munich.
Madrid was without veteran Sergio Ramos after he joined Paris Saint-Germain, with Nacho Fernandez playing in central defense when needed.
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