The shooting occurred about 4:50 p.m. on Hope Street between 11th and 12th streets, according to LAPD Capt. Andrew Neiman.
Officers from the LAPD's Central Traffic Division were flagged down by an L.A. City Department of Transportation officer and directed toward the man, Neiman said. The man was wielding a large kitchen knife, he added.
The man was struck by gunfire from at least one officer, and he was taken to a hospital, where he was pronounced dead, Neiman said.
No officers were injured. The officers were wearing body cameras at the time of the shooting, Neiman said.
As police investigated the incident along Hope Street, which was closed to traffic between 11th and 12th streets, a woman arrived at the scene and notified officers of a dead woman in a nearby apartment.
Officers went to an apartment complex in the 1100 block of Flower Street, one block from the police shooting, and found a woman in her 40s who had died "of an obvious homicide," Neiman said.
The proximity of the slaying to the confrontation with the knife-wielding man raised suspicion.
"At this point, we don't know if they are connected, but we are looking at the potential for that," Neiman said.
The identity of the slain woman was not released pending notification of her family. The identity of the man killed in the confrontation with police was also not released.
9:20 p.m.: This article was updated with confirmation that the man had died as well as with information on a nearby homicide investigation.
7:25 p.m.: This article was updated with comments from LAPD Capt. Andrew Neiman.