For those who don't know Karatsuba's Method for Multiplying Numbers, here is a Wiki here:
https://en.wikipedia.org/wiki/Karatsuba_algorithm
If you have Alfred Cromwell's Higher Algebra book, you will know that you can multiply certain polynomials using mainly the coefficients. The question though is - is it possible to use something similar to Karatsuba's method to multiply polynomials' coefficients to add and multiply more larger complex polynomials more quickly? Maybe something can be done for matrices as well?