The bodies of Kevin Armstrong, 53, and Pauline Cockburn, 48, were discovered at a property in Ladyrig View in the village of Heiton, near Kelso, at around 3pm on December 26.
Post-mortem examinations have now been carried out on both bodies.
Ms Cockburn’s death is being treated as murder while Mr Armstrong’s death is not being treated as suspicious, police said on Saturday.
They are not looking for anyone else in connection with the murder of Ms Cockburn, from Kelso, who has now been formally identified.
Detective Chief Inspector Keith Hardie, of the Major Investigation Team, who is leading the investigation, said: “Pauline was tragically taken from her family and friends at what should be a happy and joyous time of year.
“I can only imagine that the circumstances make this even more heartbreaking and my thoughts are with them as they try to come to terms with their loss.
“Our investigation into her death is ongoing and, whilst we are not looking for anyone else, there are still lines of inquiry to follow up to ensure we know as much as we can about what happened.
“This incident has put a small, quiet area of the Scottish Borders in the spotlight and I would respectfully request that the families and friends of those involved, and the local communities, can be given the time and the space to be able to grieve and remember.”