Matt released a statement through his Twitter alongside a picture of his late wife, reports variety.com.
"Halyna inspired us all with her passion and vision, and her legacy is too meaningful to encapsulate in words," he wrote.
"Our loss is enormous, and we ask that the media please respect my family's privacy as we process our grief. We thank everyone for sharing images and stories of her life."
On Thursday, while Halyna was in New Mexico shooting the western 'Rust;, the cinematographer was shot by a prop gun containing a live round of ammunition.
She was transported via helicopter to the University of New Mexico Hospital in Albuquerque, and died shortly after. Authorities have confirmed that veteran star Alec Baldwin discharged the shot that killed Halyna, though an ongoing "shooting investigation" is continuing as to determine key details in the lead-up to the accident.
Following her death, colleagues and friends in the industry offered remembrances of the cinematographer.
"We lost a truly unique artist, and it pains me to think of the images she never had a chance to create," said fellow cinematographer Stas Bondarenko.
Halyna was selected a rising star by American Cinematographer magazine in 2019 and was making a name for herself on productions such as 'Archenemy', 'Snowbound', 'Darlin', 'Blindfire' and 'The Mad Hatter'.