The beginning of the PEP (linked document) has a rationale for it, including an argument that matrix multiplication has a combination of features which distinguish it from other binary operations. Check it out. Like most Python PEPs it is quite readable. There are even alternative proposals and links to where the discussion of this issue has occurred.

It seems that the most fundamental reason is that matrix and memberwise multiplication are two distinct operations which use the same datatypes as operands and both have a very good claim to use of the * operator.

Also, just because a language is general purpose does not mean that it cannot support features which might be restricted in their use.

Two of the python's most popular uses are in scientific and numeric fields (see SciPy, NumPy and ipython). This operator seems like it would be very useful for improving readability in those areas which is probably reason enough to use it.

However, please resist the urge to just overload this new operator for something unrelated that you want. The point of infix operators is to improve the readability of code and lessen the cognitive load on the reader. Overloading an operator for a purpose which is not conventional goes exactly against this for... what benefit exactly?