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It is based on a way of multiplying two 2 × 2-matrices which require only 7 multiplications (instead of the usual 8), at the expense of several additional addition and subtraction operations. Applying this recursively gives an algorithm with a multiplicative cost of O ( n log 2 7 ) ≈ O ( n 2.807 ) {\displaystyle O(n^{\log _{2}7})\approx ...
The key observation is that multiplying two 2 × 2 matrices can be done with only 7 multiplications, instead of the usual 8 (at the expense of 11 additional addition and subtraction operations). This means that, treating the input n × n matrices as block 2 × 2 matrices, the task of multiplying n × n matrices can be reduced to 7 subproblems ...
The following exposition of the algorithm assumes that all of these matrices have sizes that are powers of two (i.e., ,, ()), but this is only conceptually necessary — if the matrices , are not of type , the "missing" rows and columns can be filled with zeros to obtain matrices with sizes of powers of two — though real implementations ...
For example, if A, B and C are matrices of respective sizes 10×30, 30×5, 5×60, computing (AB)C needs 10×30×5 + 10×5×60 = 4,500 multiplications, while computing A(BC) needs 30×5×60 + 10×30×60 = 27,000 multiplications. Algorithms have been designed for choosing the best order of products; see Matrix chain multiplication.
The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions. In mathematics, the Hadamard product (also known as the element-wise product, entrywise product [1]: ch. 5 or Schur product [2]) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.
The product of two quaternionic matrices A and B also follows the usual definition for matrix multiplication. For it to be defined, the number of columns of A must equal the number of rows of B . Then the entry in the i th row and j th column of the product is the dot product of the i th row of the first matrix with the j th column of the ...
Take the sequence of matrices and separate it into two subsequences. Find the minimum cost of multiplying out each subsequence. Add these costs together, and add in the cost of multiplying the two result matrices. Do this for each possible position at which the sequence of matrices can be split, and take the minimum over all of them.
The operation is a component-wise inner product of two matrices as though they are vectors, and satisfies the axioms for an inner product. The two matrices must have the same dimension—same number of rows and columns—but are not restricted to be square matrices .