The Czech forward spent last season on loan in England but he played in the Championship for Udinese aligned side Watford where the 21 year-old scored 22 goals.
"I'm delighted to welcome Matej to the club,' said Clarke on the official club site.
"He's had a really good year with Watford and we believe that he's got the quality to score goals at a higher level. Hopefully, he'll show that with us this year."
"We had Matej watched throughout the season. As well as some excellent scouting reports, we had some really good feedback from within Watford about Matej as a player and also a person."
Vydra completed a medical at the Baggies on Tuesday morning and now could join the team for a friendly to be held in Hungary before possibly making his debut for the club against Southampton this weekend.