Matri, 29, is believed to have signed on a four-year deal at San Siro, rejoining the side he played for as a youngster before having spells at Cagliari and Juventus, in a deal reportedly worth €11 million.
"It's a great feeling to come back to where you played as a boy," Matri told the club's official website.
"I know a lot of people here and the coach is someone I’ve worked with.
"There are many positive aspects. I spoke to [Massimiliano] Allegri and it’s up to me to get a place on the starting line-up but I’m ready to play.
"There was some interest from other clubs but as soon as Milan came in for me, I only wanted to come here. It was an objective for me and now it'll be a great starting point."The striker fell down the pecking order at the Bianconeri, with other attackers Fernando Llorente and Carlos Tevez joining Antonio Conte's ranks earlier this summer.
But, with Milan securing their place in the Champions League proper, having seen off the challenge of PSV, and Giampaolo Pazzini's return still to be confirmed, coach Massimiliano Allegri has bolstered his squad with the capture of the forward.
Matri made 83 appearances for Juventus, scoring 29 goals.
Meanwhile, Kevin-Prince Boateng has departed the Stadio Giuseppe Meazza, completing a move to Bundesliga club Schalke on Friday.