Question

Matrix multiplication of rectangular matrix

Answer 1

Let α be the maximal value such that the product of an n × nα matrix by an nα × n matrix can be computed with n2+o(1) arithmetic operations. In this paper we show that α >; 0.30298, which improves the previous record α >; 0.29462 by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an n × nk matrix by an nk × n matrix, for any value k ≠ 1. The complexity of this algorithm is better than all known algorithms for rectangular matrix multiplication. In the case of square matrix multiplication (i.e., for k = 1), we recover exactly the complexity of the algorithm by Coppersmith and Winograd (Journal of Symbolic Computation, 1990). These new upper bounds can be used to improve the time complexity of several known algorithms that rely on rectangular matrix multiplication. For example, we directly obtain a O(n2.5302)-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, where n denotes the number of vertices, and also improve the time complexity of sparse square matrix multiplication

Answer 2

Step-by-step explanation:

diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2-by-2 diagonal matrix is , while an example of a 3-by-3 diagonal matrix is.

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