Deputies picked up the man, described as a local transient, on Friday after spotting him in surveillance footage taken near the crime scene.
Authorities said he has since been released from custody, but remains a person of interest in the case.
"He was transported to a suitable facility where he is receiving the appropriate level of care. No one has been named a suspect at this point, and no arrests for the crime have been made," the Los Angeles County Sheriff's Department said in a press release.
Susan Leeds, 66, was found stabbed to death inside her car at the Promenade on the Peninsula shopping center parking lot. She was initially located by other shoppers. By the time help arrived, she was already dead, stabbed several times in her upper body.
A motive behind the killing is not known.
Sheriff's officials as well as the city manager were in a small room answering questions from citizens about the investigation.
"The Facebook Live is a way for us to get as much information as we can to the public in one quick setting," City Manager Doug Prichard said.
Anyone with information about this incident is urged to contact the L.A. County Sheriff's Department's Homicide Bureau at (323) 890-5500. If you prefer to provide information anonymously, you may call Crime Stoppers by dialing (800) 222-TIPS (8477).