Products of Matrices, Expressed as Summations

For the simple 2x2 matrices used above the product is quite easy to write out in fiiU. However, as the number of basis functions increases, it can become cumbersome to write out the full matrix and we may prefer to write out formulae for obtaining a general element of the product matrix. Products of matrices are obtained using the multiplication procedure of taking rows from the first matrix with columns from the second. To extend this to the 3x3 case and beyond, we will look at the general matrix product  

A and B are any two 3x3 matrices whose elements are denoted a and b with subscripts giving the row and column of the element in that order. The product of A and fi is the 3 X 3 matrix P. To generate the product, each column of the matrix B is treated like the vectors in the earlier calculations, being multiplied element by element with a row from the A matrix and then summed. Each row of the A matrix is used in turn, so that the three rows of A and the three columns of B form the nine sums of elemental products needed to generate a 3 x 3 matrix as the result. The product matrix elements are labelled by the row of A and column of B used in the multiplication. For example, element 1,2 of the result pu is found by multiplying each element of the second column in matrix B 

Similarly, element is calculated from row 3 of matrix A and column 1 of matrix B  

In this way, each of the nine elements of the product can be built from the elements of the A and B matrices. The sums that are written out for the particular cases shown in Equations (A5.8) and (A5.9) have an index k which defines the column index of the a element and the row index of the b element used in each term of the sum. This index disappears in the final answer for the product element produced and only the fixed indices remain to define p. For an arbitrary product element Py we can use the formula 

But we can also use the matrix product to generate a matrix for the operation and then apply this as a single operation, giving the same result  

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