The 29-year-old has been a key member of the squad throughout the club’s time in the Premier League and for three-and-a-half seasons was the club’s first choice keeper.
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Head coach Graham Potter paid tribute saying, “Mat has been a great servant to the Albion over the past four years and on behalf of the club I’d like to place on record our gratitude for his efforts.
“He’s a great guy, top professional and as someone who wants to be playing regularly at senior level he goes to Real Sociedad with our very best wishes.
“He’s been a pleasure to work with, he will be fondly remembered by everyone and always welcome back at the club.”
Ryan joined on a five-year deal from Valencia in the summer of 2017, for a then club-record transfer fee.
Having played for Blacktown City and Central Coast Mariners in Australia between 2009 and 2013, Ryan moved to Europe, signing for Club Brugge where he made more than 100 appearances for the Belgian Pro League team.
After winning the Belgian Cup in 2015 and Belgian Pro League Goalkeeper of the Year in both 2014 and 2015, the shot-stopper joined Valencia in 2015 and signed a six-year deal, keeping a clean sheet on his league debut against Rayo Vallecano.
Ryan made his international debut for Australia in 2012, and played all three matches in the 2014 World Cup in Brazil, before winning the AFC Asian Cup with the Socceroos in 2015.
He made his Seagulls debut in their opening Premier League fixture against Manchester City at the Amex, and went on to play every minute of top-flight football for the club during the 2017/18 campaign.
The Australian continued his fine form throughout the 2018/19 season, and was named the PFA Bristol Street Motors Player of the Month for October.
He also represented his country at the 2019 Asian Cup, where Australia reached the quarter finals, losing 1-0 to the United Arab Emirates.
The 28-year-old continued as first choice goalkeeper under Graham Potter, keeping nine clean sheets in 38 league appearances in the 2019/20 campaign.
Having made 11 appearances in the first half of the 2020/21 Premier League campaign, Ryan joined Arsenal on loan for the remainder of the season, making three appearances for the Gunners.