The local minister said the death of Eleanor Whitelaw was a "distressing tragedy" as flowers were laid at her door.
The 85-year-old died in Edinburgh Royal Infirmary on Monday night. She was being taken there on July 11 after she was found injured at the flat in Morningside Grove in Morningside.
Officers said the incident is now being treated as murder.
The attack took place while Mrs Whitelaw's husband was at the shops. She was on her own in the house for around an hour and a half and was found in the living room on his return.
The Rev Dr Derek Browning, minister of Morningside Parish Church, yesterday spoke of the effect on the family and community of the death. The congregation had said prayers for her since she was taken to hospital.
He said: "When distressing tragedies like these happen - in any community - our first thoughts should be with the family that has been directly affected.
"I am sure the thoughts and prayers of everyone in Morningside are with the Whitelaw family today. The community remains concerned and upset about what has happened."
Mrs Whitelaw's son is Professor Bruce Whitelaw of the Roslin Institute, Edinburgh, where Dolly the cloned sheep was created in 1996.
Mr Whitelaw is deputy director and head of division of developmental biology.
Residents who live near the couple paid tribute to Mrs Whitelaw, who they described as sparkling but frail.
Douglas McPhie, 90, and his wife 93, knew Mrs Whitelaw. He said: "A lot of people round here knew her because their house was right at the bus stop.
"She was a sparkling person.
"It's terribly sad, but we are most concerned about the family. We have known them for about 30 years. She was getting frail. But she was a good neighbour."
Robert Buczek, 23, was charged with attempted murder and assault to severe injury and danger of life when he appeared at Edinburgh Sheriff Court on Monday in connection with the case.
Buczek made no plea or declaration and was remanded in custody after the appearance in private.
Confirming that the case was now a murder investigation, Police Scotland said: "A 23-year-old man has made a first appearance at Edinburgh Sheriff Court and is remanded in custody."
l A 90-year-old woman is being treated in hospital after she was allegedly raped in her own home by a teenager. She was taken to hospital on Sunday morning.
Kriss Strachan, 17, of Bellshill, Lanarkshire, has appeared at Hamilton Sheriff Court charged with rape. The apprentice car mechanic was arrested by police after they were called to a house in Bellshill.
He appeared in private and made no plea before being remanded. He will appear in court again in a week's time.