Vincente David Montano was committed twice in 2004 and twice in 2007, said Aaron, citing officials in Rutherford County. Montano had been arrested Murfreesboro, Tennessee, in 2004 in a case of assault and resisting arrest, police said.
"We have no motive for (Wednesday's attack)," Aaron said.
Montano had an airsoft pistol with him that he aimed and fired at police in the theater, Aaron said. Such a weapon looks like a semiautomatic pistol but fires plastic or BB pellets.
Montano's mother filed a missing person's report with Texas Rangers two days ago and they had notified authorities in Tennessee, Aaron said.
In the report, his mother, Denise Pruitt, told authorities that Montano was diagnosed with paranoid schizophrenia in 2006.
"Ms. Pruitt advised Vincente has several other health issues and has a hard time taking care of himself," the report says. It lists his address as "homeless."
Pruitt, who lives in Florida, last saw her son in Illinois in March 2013.
Aaron told reporters that one patron at the screening of "Mad Max: Fury Road" suffered a minor cut on a shoulder from a hatchet before officers killed the suspect.
Police Chief Steve Anderson said that Montano was 29 years old. He had an identification card that listed a Nashville address, Anderson said, but authorities want to check fingerprints to confirm his identity.
The wounded man had a superficial injury to his arm, fire department spokesman Brian Haas said.
The man, who was identified as Steven, told reporters that police did a "phenomenal job" in responding to the calls for help. He was one of eight people in the theater, and one of three who were pepper sprayed by the assailant. He said his daughter was also sprayed.
"I am very, very grateful that no one else got injured here today -- other than the person who perpetrated this," Steven said.
No one was transported to a hospital, Haas said, but Steven, his daughter and another woman suffered irritation from pepper spray apparently used by the suspect, who was wearing a surgical mask. The pepper spray contained red dye, Anderson said.
The suspect had two bags -- a backpack and a satchel -- with him, Aaron said. The bomb squad destroyed the backpack, which contained a "hoax explosive device," Anderson told reporters.
Police were called to the Carmike Hickory 8 movie theater complex in the Nashville suburb of Antioch at 1:13 p.m. CT. Two officers were already at the mall working a car accident when people came running toward them. The officers were inside the complex withing two minutes, Aaron said.
One officer -- who entered the theater and fired at the man before backing away -- said Montano pointed the airsoft pistol at him and pulled the trigger.
The SWAT team arrived and had to don gas masks due to the heavy chemical spray in the theater, Aaron said. The man tried to flee out the back door.
Brad Ransom, who told CNN he works across the street from the theater, said he saw four or five officers along the side of the building around the time of the fatal shooting.
More than a dozen gunshots, fired in rapid succession, can be heard in a video recorded during the fatal shooting. The video doesn't show the shooting.
A woman who worked at a Sprint store near the scene told CNN's Brooke Baldwin that about three hours earlier, a man with two backpacks tried to enter her store through their back door.
The Bureau of Alcohol, Tobacco, Firearms and Explosives sent a team to help with the investigation.
The FBI and Tennessee Bureau of Investigation were assisting.
Antioch is a community in southeast Nashville.