The Stillwater actor remembered a promise he'd made to his former pal when his spouse, Luciana Barroso, "announced" they were getting inked in 2013 and contacted Scott Campbell, who had been responsible for the 'Brokeback Mountain' actor's body art to carry out the work.
He recalled: "She just announced it. We were in our apartment in Manhattan, and she was, like, 'We're getting tattoos.' I was, 'Okay.'
"There is a friend of ours who did all of Heath Ledger's tattoos, and I told him if I ever got a tattoo, he was my first phone call."
Scott came to the couple's house and freehanded the name Lucy on Matt's arm in honour of his wife, and the 50-year-old star - who has Isabella, 15, Gia, 12, and Stella, 10, with his wife, and is also stepfather to 22-year-old Alexia - decided to get another inking, a loopy line heading up towards his shoulder, to match one which Heath had.
Asked about the unusual inking - which is above the name of his wife and kids - he explained to America's GQ magazine: "That's something that Heath had on his arm. Heath was an incredibly restless, creative person.
"Like, I talked to the person who did his hair on The Patriot and she said he hated sitting still so much 'that by the time I got the wig on and I set it and everything, and I'd finished, he'd get up and there would be a sculpture of bobby pins that he'd done.'
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"He was really sensitive. This stuff just flowed out of him. He was really special.
"I just wanted to get something that Heath had. Scott showed me his laptop and I said, 'Scott, what's that one?' And he goes 'I have no idea - I think that's just some s*** that Heath squiggled.' And I went, 'That's the one I want.'"
Luciana was also friends with Heath and had the same inking etched on her foot.
Matt added: "So we both have that. It's like a little creative little blessing. It's like an angel that looks over all these names that are on the arm."