Gunman infatuated with extremism, mentally unstable - Australian PM

"He had a long history of violent crime, infatuation with extremism and mental instability," Abbott said. "As the siege unfolded ... he sought to cloak his actions with the symbolism of the ISIL death cult."

The 50-year-old, widely named in the media as Man Haron Monis, took 17 people hostage at the Lindt cafe in central Sydney on Monday.

He was killed when police in SWAT-style gear stormed the eatery early Tuesday. Two hostages also died.

Abbott said he was well known to Australian authorities.

"We know that he sent offensive letters to the families of Australian soldiers killed in Afghanistan and was found guilty of offences related to this," he said.

"We also know that he posted graphic extremist material online. Tragically, there are people in our community ready to engage in politically motivated violence."

Abbott praised police for the way they acted, saying Australians "should be reassured by the way our law enforcement and security agencies responded to this brush with terrorism".

"Plainly, there are lessons to be learned and we will thoroughly examine this incident to decide what lessons can be learned," he said, adding that "it will take time to clarify exactly what happened in Martin Place and why".