The graphic film shows a violent struggle between the man and several officers in the city’s Skid Row area.
As the police try to subdue him, a voice is heard shouting “Drop the gun!” Seconds later, five gunshots ring out.
Police have yet to confirm whether the man was armed.
Skid Row has a large homeless population.
Police spokesman Sgt Barry Montgomery told the LA Times newspaper that officers had been responding to reports of a robbery in the area.
He declined to say how many officers had been involved in the incident but confirmed that none had been injured. He did not identify the dead man.
The police department said officers had attempted to use a Taser to subdue the suspect but he had “continued fighting and resisting”.
LA police commander Andrew Smith said that at some point there had been a struggle over one of the officer’s weapons.
No other gun was recovered at the scene, he told the LA Times.
Police would have been allowed to use deadly force if the man had a weapon.
Five shots
It is unclear from the video what exactly happened, the BBC’s Alastair Leithead reports from LA.
The video begins with a man swinging punches at four police officers on a pavement in daylight. Two other officers run up to join their colleagues.
As the man is knocked to the ground, a woman can be seen picking up a fallen police baton and a voice appears to shout “Give me my stick! Give me my stick!”
Several police officers turn to tackle the woman, in the foreground of the video, and her detention partially obscures what is happening with the man on the ground.
A voice can be heard shouting “Drop the gun!” four times before five shots ring out.
The police officers step away from the suspect lying on the ground and nobody approaches the body again for one minute 13 seconds, by which time police reinforcements have arrived.
The man taking the video can be heard swearing as he records it.
After the shooting, he can be heard saying. “Ain’t nobody got no [obscenity] gun. I’m gonna record this [obscenity]… They just shot that man right here, man, they just shot that [obscenity] man right here, yeah.”
Tensions
The president of LA’s Police Commission, Steve Soboroff, told the BBC it was important to investigate the facts before making any conclusions.
“I would just encourage and pray that people not rush to judgement,” he said. “It isn’t fair to anyone. It’s not fair to the family of the victim, to the victim’s memory or to the officers,”
Last year, highly publicised killings of black men sparked weeks of protests in the US.
The shooting dead of Michael Brown – an unarmed black 18-year-old – by a police officer in Ferguson, Missouri, last August sparked protests across the US.
Tensions were further inflamed in January when police in New Jersey were caught on camera shooting dead a black man as he stepped out of a car with his hands raised.
Source: BBC