Multiplication of matrix

Hie multiplication of matrices requires a bit more reflection. The product C of two matrices A and B is usually defined by C = AB if... 

First we present the rules for equality, addition, and multiplication of matrices. Equality... 

The multiplication of matrices, however, is generally not commutative i.e. the order of the factors must not be changed. 

This modification for Boolean matrix multiplication permits use of the Boolean union operation (logical OR operation or logical sum) instead of regular multiplication and union operations. The Boolean union operation can be executed much faster on a digital computer. Experience has shown that performing the Boolean union of rows instead of the standard Boolean multiplication of matrices can reduce the computation time by as much as a factor of four. 

Direct multiplication of matrices, which are subscripted with composite indices, is carried out in the same manner as that of doubly-subscripted matrices e.g. the direct product in equation (93a) can be expressed as ... 

Multiplication of matrices is not commutative, that is A.B B.A even if the second product is allowable. Matrix multiplication can be expressed as summations. For arrays with more than two dimensions (e.g. tensors), conventional symbolism can be awkward and it is probably easier to think in terms of summations. 

This empirical multiplication rule was soon identified as equivalent to the mathematical rule for the multiplication of matrices, with two important implications ... 

Multiplication of matrices is possible only when the number of columns in one matrix is equal to the number of rows in the other, since it is performed by multiplying the elements of the ith row in B by the corresponding elements of the yth column in A, and summing the products to give the element in the /th row and yth column in the product BA. If... 

For example in clock arithmetic all the basic laws hold except for cancellation. In clock arithmetic, 3X4 = 3X8 because both leave the hands in the same position, but of course, 4 does not equal 8. In the multiplication of matrices, commutativity does not hold. 

You may have observed from your answers to Problem 4.4 that multiplication of matrices follows similar rules to that of numbers, insofar as it is ... 

When deciding whether a given set of matrices of order n forms a group under multiplication, we can disregard associativity as one of the criteria because multiplication of matrices is always necessarily associative - and so we only need to check for closure, the presence of E , and identify all inverses. 

To do this we have to form the squares and This is carried out according to the rule given in the text (p. 117) for the multiplication of matrices. If a and b are two matrices, the elements of their product c= ab are given by v... 

The algebra of multi-way arrays is described in a field of mathematics called tensor analysis, which is an extension and generalization of matrix algebra. A zero-order tensor is a scalar a first-order tensor is a vector a second-order tensor is a matrix a third-order tensor is a three-way array a fourth-order tensor is a four-way array and so on. The notions of addition, subtraction and multiplication of matrices can be generalized to multi-way arrays. This is shown in the following sections [Borisenko Tarapov 1968, Budiansky 1974],... 

The non-commutative nature of the multiplication of matrices is of great importance in matrix mechanics. The difference of the product of the matrix q,- representing the coordinate gr, and the matrix p, representing the canonically conjugate momen-... 

The most dramatic contrast between multiplication of matrices and multiplication of numbers is that matrix multiplication can be noncommutative, that is, it is not necessarily true that... 

The sum is over all of the Lj,- values found within the period length. Operationally we can obtain Tr Gi2--.3i and d Tr Gi2..-3i from (2.33) through numerical matrix multiplication. The extension to a DNA of period / is made by replacing 6 with / in (2.36) and carrying out the appropriate multiplications of matrices. 

The proposed optimisation algorithm enables one to find local optimum of a general objective function. The evaluation of objective functions requires only a modest amount of computation (in the essence the multiplication of matrices Z, 2 and Z, 4b by vectors of variances of measured quantities). Even problems of realistic dimensionality can be solved efficiently on personal computers. 

When working with MATLAB, it should be noted that there is also an element-by element multiplication for matrices that is completely different from ordinaiy multiplication of matrices described above. The element-by-element multiplication, whose operator is (a dot before the ordinary multiplication operator) may be applied only to matrices of the same order, and it simply multiplies corresponding elements of the two matrices. For example, if A and B are of order 3 x 2), then the eleraent-by-element product A. B is... 

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