what matrix operations have better known time complexity on a quantum computer?

ANSWER MADE CW

Arguably the one thing that a quantum computer can do quickly, as long as the matrix is defined in the appropriate manner, is the quantum Fourier transform (QFT). Much as the classical fast Fourier transform (FFT) is the workhouse of many classical matrix algorithms, the QFT is the basis for many such quantum matrix algorithms.

• For example, a standard answer about quantum algorithms for matrices is the quantum algorithm for linear systems - the HHL algorithm of Harrow, Hassidm, and Lloyd. This algorithm allows one to invert a matrix.

• Another interesting matrix algorithm is that of Janzing and Wocjan on determining the elements of a power of a matrix.

• One algorithm that I haven't studied in detail but nonetheless seems interesting is Wang's method of determining the effective resistance between two nodes in a graph - if the network were considered to be an adjacency matrix of resistors.

Without knowing more about the OP's background, at least hearing of the importance of the QFT and its embodiment for quantum phase estimation (QPE) is critical to understand how quantum computers can be useful for matrix algorithms.