The three Michigan State students killed in the shooting have been identified by university police.

Brian Fraser, a sophomore, and Arielle Anderson, a junior, were both from Grosse Pointe, Michigan.

Alexandria Verner, a junior, was from Clawson, Michigan.

Five other students remain in the hospital in critical condition following the attack.

The FBI offered new details on the suspected gunman in a confidential briefing to law enforcement on Tuesday.

Investigators have determined 43-year-old Anthony McRae walked from his home to Michigan State and had no connection to the university or the victims.

Authorities said that when McRae was found dead from a self-inflicted gunshot wound, he had a “3-page document expressing his reasons for the attack and a number of additional locations in Lansing and Holt, Mich.; Ewing Township and Franklin Park, NJ; and Colorado Springs, Colo., which had ‘hurted’ (sic) him and, therefore, were deserving of attack.” Agents believe he had “personal grievances” with people at those locations.

McRae lived with his father, who is cooperating with the investigation, the FBI reported.

Investigators said the suspect’s writings confirmed he “was often alone.” The briefing said the gunman viewed himself as “a loner” and an “outcast” who was “never noticed or accepted by others.”

-ABC News’ Josh Margolin

President Joe Biden said in a statement Tuesday, “Our hearts are with these young victims and their families, the broader East Lansing and Lansing communities, and all Americans across the country grieving as the result of gun violence.”

Biden noted that the Michigan State mass shooting came one night before the U.S. marked five years since the Parkland, Florida, high school massacre. In the Feb. 14, 2018 school shooting in Parkland, 17 students and staff were gunned down.

"I have taken action to combat this epidemic in America, including a historic number of executive actions and the first significant gun safety law in nearly 30 years, but we must do more,” Biden said.

He stressed, “Congress must do something and enact commonsense gun law reforms, including requiring background checks on all gun sales, banning assault weapons and high-capacity magazines, closing loopholes in our background check system, requiring safe storage of guns, and eliminating immunity for gun manufacturers who knowingly put weapons of war on our streets. Action is what we owe to those grieving today in Michigan and across America.”

The first active shooter call came in at 8:18 p.m. from Berkey Hall, an academic building, and the university immediately told students to shelter in place, the Michigan State University Police Department said.

Shots were fired soon after at the student union.

At about 11 p.m., the suspected gunman was seen on campus security cameras, police said.

Images of the suspect were shared with the public at 11:18 p.m., police said.

At about 11:35 p.m., a caller’s tip led police to the suspect, authorities said.

Hundreds of people gathered at memorials, prayer services and candlelight vigils across Michigan on Tuesday night to honor the three students who were killed and five others who were injured in Monday's mass shooting at Michigan State University.

More vigils are planned for the rest of the week, including one scheduled for Wednesday at 6 p.m. ET at The Rock, a 149-year-old landmark on MSU's main campus in East Lansing.