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That is, the entry of the product is obtained by multiplying term-by-term the entries of the i th row of A and the j th column of B, and summing these n products. In other words, is the dot product of the i th row of A and the j th column of B.
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
Order of operations. In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression.
According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain. [ 18 ] [ 19 ] Other authors define a principal submatrix as one in which the first k rows and columns, for some number k , are the ones that remain; [ 20 ] this type of submatrix has ...
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.
The formula is valid for all index values, and for any n (when n = 0 or n = 1, this is the empty product). However, computing the formula above naively has a time complexity of O( n 2 ) , whereas the sign can be computed from the parity of the permutation from its disjoint cycles in only O( n log( n )) cost.
In general, indices can range over any indexing set, including an infinite set. This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation. An index that is summed over is a summation index, in this case "i ".
It is common convention to use greek indices when writing expressions involving tensors in Minkowski space, while Latin indices are reserved for Euclidean space. Well-formulated expressions are constrained by the rules of Einstein summation : any index may appear at most twice and furthermore a raised index must contract with a lowered index.