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3.4 Matrix multiplication. 3.5 ... is a notational convention that implies summation over a set of indexed terms in a formula, ... Einstein notation can be applied in ...
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]
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:
Gerhard Kowalewski and Cuthbert Edmund Cullis [82] [83] [84] introduced and helped standardized matrices notation, and parenthetical matrix and box matrix notation, respectively. Albert Einstein (1921) Albert Einstein, in 1916, introduced Einstein notation, which summed over a set of indexed terms in a formula
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions ...
While some relativists consider the notation to be somewhat old-fashioned, many readily switch between this and the alternative notation: [1] = The metric tensor is commonly written as a 4×4 matrix. This matrix is symmetric and thus has 10 independent components.
That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. The components of vectors (as opposed to those of covectors) are said to be contravariant. In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices as in