Bill Belichick Autographed Jordon Hudson's School Work During 1st Meetup

During Bill Belichick's first interaction with his now-girlfriend Jordon Hudson, he actually gave her an autograph -- one in which he thanked her for an impromptu lesson on logic.

Check out images of the signature that TMZ Sports obtained from one of Hudson's ex-Bridgewater State University classmates. You can see the former New England Patriots head coach wrote the then-21-year-old a sweet message on the inside of a textbook titled "Deductive Logic" -- expressing his gratitude to her for "a course" on the subject -- before he slapped his John Hancock on it.

Our sources say Belichick put the pen to the paper during the two's first meetup on a February 2021 plane ride from the Boston area to Florida -- after he had leaned in to inquire about the matter.

We're told Belichick chatted with Hudson about logic for a while -- before introducing himself, and later signing the book. Our sources say the duo then swapped contact information.

You can see in addition to the cool note, Belichick also told her to travel safe ... while adding the years of each of his Super Bowl wins.

As we reported, the former college cheerleader and the now-72-year-old Belichick stayed in contact following the run-in ... bonding over the philosophy talk and more. Eventually, after Belichick's split with Linda Holliday, they became romantic.

Neither Belichick nor Hudson has publicly commented on the relationship ... though at Tom Brady's Patriots Hall of Fame induction ceremony last week, 24-year-old Hudson wrote on a photo of the future Hall of Famer, "You are the embodiment of virtue, dedication, humility and grace. I am deeply moved by you."