Matrix multiplication with the standard algorithm has an algebraic complexity of O(n^3) for n x n matrices, but in 1969 Strassen found another algorithm for multiplying 2 x 2 matrices with which he showed that matrix multiplication can be done with a complexity of O(n^2.81) by applying his algorithm recursively for large matrices. We present a historical overview of the best known bounds on the matrix multiplication exponent, with the current best known bound of 2.371866 and methods, used to find new algorithms for other matrix multiplications, such as alternating least squares (ALS) and SAT solving. After this we present a novel method with which we found a rank 23 decomposition of the <3,3,3> matrix multiplication tensor that may be inequivalent to known decompositions. We also found decompositions for <2,2,2>, <2,2,3>, <2,2,4>, <2,3,3> and <2,2,5> that provide the same bounds as earlier found decompositions. Our method found many decompositions for the smaller tensors but only one for the larger due to time limits. The method is based on the alternating least squares method (ALS) with the modification that we 'push' the coefficients towards integer (or rational) numbers. After a number of iterations and rounding the result this sometimes yields exact decompositions so that we can bound the matrix multiplication exponent. ... Read more