By Caroline Wyatt
BBC News defence correspondent
Sadr City is a test of the Iraqi government's authority
A fragile peace appears to be holding in Sadr City, the Shia stronghold of the anti-American cleric Moqtada Sadr, despite the killing of 11 people in a gun-battle that followed a roadside bombing in a nearby area of eastern Baghdad.
The Iraqi army went in to Sadr City in their thousands on Tuesday: on foot, in tanks and in armoured humvees, taking up sentry positions on rooftops and at crossroads as soldiers patrolled the streets and began to defuse roadside bombs laid by the Mehdi Army, the Shia militia.
The Iraqi forces' entry into the poor Shia area, home to around two million people, came after a ceasefire between the Iraqi government and the Sadrists was agreed on 10 May, coming into effect soon afterwards.
The Shia militiamen agreed to put down their arms - though not to give them up - as long as the patrols were limited to Iraqi security forces, with the US military staying out of Sadr City.
The deal came after weeks of heavy clashes between the US-backed Iraqi army and the militia, in which
hundreds of people - many of them civilians - died, and up to 2,000 were injured.
Some shops have now re-opened, and civilians on the streets seemed relieved that the ceasefire was holding.
One man told journalists: "We want nothing but security and stability. Inshallah, God will keep the Iraqi army safe. We will co-operate with them."
Another, a woman in her 40s, said: "We want electricity and water, but above all we want security. Our houses are destroyed and our boys have no jobs."
'People were suffering'
So why are Moqtada Sadr's men co-operating now?
Iraqi troops entered the outskirts of Sadr City on Saturday
His spokesman, Salah Al Ubaidi, says that they hoped to achieve many things from the ceasefire.
"We want the situation to calm down. On our side, we have worked hard towards that, taking many steps to make Iraq calmer. It was a good message to the Sunnis, who kicked out al-Qaeda from their side," he told the BBC in an interview.
"It was a message to them to work against al-Qaeda and understand that we are brothers and citizens of the same country.
"At the same time, freezing Jaysh Al Mehdi (the Mehdi Army) was a message to the Iraqi government to work reasonably, and to help make the Iraqi people respect each other without using force.
"The step towards ending the fighting in Sadr City came as part of our strategy because our people were suffering too much."
However, he re-iterated that Moqtada Sadr's supporters had no intention of giving up their weapons until the multi-national US-backed forces left the country.
Mizher Shaher Nesaif, the commander of the Iraqi Infantry Division said the Iraqi army had been welcomed by the people of Sadr City, and that its main job now would be to arrest any wanted men, help provide public services and ensure security.
The American military said the Iraqi troop deployment was planned and executed by the Iraqi army itself.
"The operation is to protect the people while setting conditions for sustainable security and increased humanitarian assistance, economic growth and essential services," according to a US statement.
But much will depend on the reactions of the fighters of the Mehdi Army. They could resist when Iraqi troops carry out their plans to confiscate any heavy weapons they find in the area, or to begin to arrest key figures.
Wall builders attacked
Much of the earlier fighting had centred around a massive concrete wall that the US military has been building to cut off at least a third of Sadr City, in an attempt to prevent the flow of weapons to the rest of Baghdad.
Mehdi Army fighters repeatedly attacked those constructing the wall under US protection.
It is not clear how that work will progress further.
All eyes will be on Sadr City in the coming days.
After exactly two years in office, Iraqi Prime Minister Nouri Maliki is determined to extend his control over areas until now ruled by the militias.
What happens in Sadr City over the next few days and weeks will prove one of the biggest tests of his government's authority so far - a test it cannot afford to fail.