The mountaineer and another Alpinist were coming back down on the northeast face when the accident happened on Sunday, the Wallis regional police said in a statement.
“At an altitude of approximately 4,000 metres, one of them fell for reasons that remain unclear,” the police said.
The other climber alerted the emergency services, but a rescue team flown in by helicopter “could only confirm the mountaineer’s death”.
“The victim is a 58-year-old South Korean citizen,” said police.
Public prosecutors have opened an investigation into the death.
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On the southern border with Italy, the pyramid-shaped Matterhorn is a cherished Swiss national icon. First climbed in 1865, the summit is 4,478 metres (14,692 feet) above sea level, making it one of the highest mountains in the Alps.
