Milan paid €11 million to sign Matri from Juventus last week.
"The last few days have been hectic and I wanted to wait a little while before writing," he posted on Facebook.
"With Juventus, I spent two intense years, beautiful, full of sacrifices, victories and friends. In football you must always be prepared for changes and new challenges and I'm ready for a new adventure.
"I send a special greeting to all of you who have supported me over the years, I know that it will be difficult for a Juventus fan to follow me again with the same passion, but I hope I have left a good memories for all of you."
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