Image caption Wade Michael Page was shot in the stomach by a police responder

The man who shot dead six people at a Sikh temple in Wisconsin died of a self-inflicted gun shot to the head, after police shot him, says the FBI.

The authorities said police responders shot Wade Michael Page, 40, in the stomach, but they did not fire the fatal shot, as they had initially said.

At a news conference, the FBI also confirmed Page's ex-girlfriend had been arrested on a weapons charge.

There is no suggestion Misty Cook, 31, was involved in Sunday's attack.

FBI Special Agent Teresa Carlson confirmed on Wednesday that she had seen video footage in which a police officer shot Page in the stomach.

"It's an amazing shot, and thank goodness," she said.

"Subsequent to that wound, it appears that Page died of a self-inflicted gunshot wound to the head," Agent Carlson added.

'No clear motive'

She said Page, a heavily tattooed Army veteran who performed in white-power bands, was still the only suspect.

Dozens of people who knew him were being interviewed, although officials have not yet "clearly defined a motive" for the shooting in a Milwaukee suburb, Agent Carlson added.

Federal officials were not investigating Page before the shooting, she said. He bought the gun used in the attack at a nearby gun shop days after he moved to Cudahy, Wisconsin.

His relationship with Misty Cook, a nursing student, ended earlier this year.

She was found to have a firearm at her Milwaukee home when police called to interview her about Page.

Image caption Another vigil was held on Tuesday night for the victims

Cook is prohibited from keeping a gun because she has a previous conviction for fleeing and eluding a traffic officer.

On Wednesday, President Barack Obama called Indian Prime Minister Manmohan Singh, who is a Sikh, to express his condolences over the tragedy, the White House said.

Hundreds of people held another vigil on Tuesday night for the victims.

Temple member Karan Singh Toor said he felt proud to be an American when he saw people of different races attending.

Other temple members have said they knew of no previous threats to the place of worship, and that Page had never been seen there before.

The dead were five men and one woman, who ranged in age from 39 to 84.

Three others injured in the attack remain in critical condition.

On Wednesday, Oak Creek Police Chief John Edwards said Lt Brian Murphy, who was shot multiple times as he tried to help a victim, was improving.

"He was up walking yesterday," Mr Edwards said. "He's progressing amazingly and we're very, very thankful for that."