The team's third-round draft pick, Mathieu did not play last season after repeated off-field issues.
The deal could be worth up to just over $3 million. According to a person familiar with the contract, Mathieu does not get his entire slotted salary bonus of $662,500 at first. Initially, he gets $265,000 with the remainder spread out over the next three seasons, as long as he does not get suspended for substance abuse.
If he's cut from the team for any other reason, he gets the entire signing bonus, said the person, who did not want to be identified because the details of contracts are not made public.
The deal includes a $170,000 workout bonus. The other figures are those normally given to someone drafted where Mathieu was.
Mathieu was a Heisman Trophy finalist in 2011. He was kicked off LSU's team before the start of last season for failed drug tests. His image and NFL future took another hit a few weeks later when he and three former teammates were arrested on marijuana charges after police officers found 10 bags of pot and drug paraphernalia in his Baton Rouge apartment.
In two seasons at LSU, Mathieu recovered eight fumbles, returning two for touchdowns, forced 11 fumbles, made four interceptions and 133 tackles, and had six sacks. He returned two punts for score.
When he appeared at his introductory news conference with the Cardinals, Mathieu said he wanted to abandon his nickname of "Honey Badger" because of all the problems associated with his time at LSU.
"I understand that there's a lot of people who love it and can't let it go," he said then, "but for me, I want people to recognize me as Tyrann. I don't have anything against the nickname, it's just that Honey Badger happened during such a dark time."
Mathieu's former LSU teammate and Arizona Cardinals standout Patrick Peterson has taken the youngster under his wing. Both are represented by the same agent.
"So grateful & blessed to be a part of Arizona Cardinals Family," Matthieu said on his Instagram account, accompanied by a photo of him smiling while he signed the deal. "This was a childhood dream of mine & now I can finally say 'I made it.'"
He is the sixth of Arizona's nine draft picks to sign a deal. Among those remaining are first round choice, North Carolina guard Jonathan Cooper, the seventh overall selection.