Matera to Positano by public transport/taxi

I cannot help with the whole route, but you can get from Matera to Naples with the FrecciaLink bus + train combination.

https://www.trenitalia.com/it/le_frecce/servizio_freccialink.html

The bus goes from Matera to Salerno, and then you take a Frecciarossa train from Salerno to Naples. The bus cannot be booked on its own; it must be connected to a reservation on a Frecce train.

The bus train link Lola posted is correct. From Salerno-Positano, use either a very expensive taxi or the cheap bus.

https://www.rome2rio.com/map/Matera/Positano

https://www.positano.com

Thank you so much to Lola and Gerri for the useful information! The Freccialink ticket is available to purchase online, so it's very convenient. But there are only 21 minutes gap to connect from the bus to the next train. And I cann't buy the earlier bus or later train seperatly to allow more connecting time. I might have to be brave enough to believe I can catch the train in time.
The rome2rio website could not find me any ticket. But thanks all the same!
Natalie

Yes, you can only buy it as one ticket. But I bet they have timed the connection to work. If it doesn’t, your train ticket is good on the next train.

Or you could just skip the train to Naples ( even though you have paid for it) and connect to Positano from Salerno. There must be a bus.

Traveling from Matera-Salerno you don't need the train (FR link). https://www.rome2rio.com/map/Matera/Salerno

From Salerno use the bus from Salerno-Amalfi
Then change bus Amalfi-Positano.

To see schedules on the Positano link above, use the drop down menu to change locations.

How big you think Salerno station is? You can tour all of it in 15 minutes and I bet most Freccia trains depart from track 1

Thank you all for the additional imput! It's so nice to receive so much help. You have all been wonderful! Thanks!
Natalie