Order, of matrix multiplication

By analogy, preserving the order of matrix multiplication, which is not commutative, the solution to the following matrix differential equation for the CSTR startup response. 

Note that the order of matrix multiplication matters. In general for two matrices A and B, AB BA. 

Note that the order of the multiplication with the square matrix diag(s) is different in the two examples. In the first example the rows are normalised, in the second, the columns. 

Binary composition in a set of abstract elements g,, whatever its nature, is always written as a multiplication and is usually referred to as multiplication whatever it actually may be. For example, if g, and g, are operators then the product g,- gy means carry out the operation implied by gy and then that implied by g,. If g, and gy are both -dimensional square matrices then g, gy is the matrix product of the two matrices g, and gy evaluated using the usual row x column law of matrix multiplication. (The properties of matrices that are made use of in this book are reviewed in Appendix Al.) Binary composition is unique but is not necessarily commutative g, g, may or may not be equal to gy gt. In order for a set of abstract elements g, to be a G, the law of binary composition must be defined and the set must possess the following four properties. 

This strategy, however, is only justified if the order of the multiplication in (2.48) is not important. This will be true if the thickness, /, of each lamella is sufficiently small. If the Jones matrix of a lamella is expanded in z about its position, z0, in the optical element, one obtains... 

Take the case of matrix multiplication, this would usually be implemented in FORTRAN as a direct translation of the expression in Table XI which is the usual "row by column and add" sequence of operations. The evaluation of a single Cjk does not satisfy the criteriof given above because although B k is a vector with unit address increment from component to component Aj, is not and its address increment is N. This of course is a function of the FORTRAN compiler and can be circumvented by storing A in an "unnatural" order (i.e. by rows, giving the transpose of A in the usual method of storage by columns) but this is not usually worthwhile because of the potential for confusion. 

Since matrix (3.46) has a block-diagonal form, its larger blocks being matrices of order 2, matrix multiplications in Equation (3.47) can be easily done block-by-block involving matrices of order 2 at most. 

Consider a square matrix [/ll of order m. Let (x) be a column matrix of the same order, that is, with m rows and 1 column. From the definition of matrix multiplication it is known that the premultiplication of the matrix (x) by the matrix [ 4] generates a new column matrix (y) so that... 

It is not difficult to come up with the matrices representing rotations about the x- and y-axes. If a vector is written as a row vector rather than a column vector, then the rules of matrix multiplication demand that the rotation matrix operates from the right in order to... 

For the different parametrization schemes of the HJfc [611], the exponential parametrization can be chosen as it requires the lowest number of matrix multiplications [646]. The number of matrix multiplications can be further reduced by two additional considerations elaborated in Ref. [647]. First, the intermediate operator products which do not contribute to the final DKH Hamiltonian can be neglected. For example, in the fcth step the matrix is multiplied to an intermediate M of order I in the potential. If fc - - 21 > n >k + l and M is even, the multiplication with Wfc can be skipped, because the intermediate term, which is the product of and M/, is odd and then does not contribute to the nth order DKH Hamiltonian. The further multiplication to yields an even matrix but goes beyond nth order. Second, the DKH Hamiltonian matrix is taken from the upper part of the transformed four-component Hamiltonian matrix while the lower part is not required. For instance, if fc + Z = n and M/ is odd, the product with the odd operator W/t contributes to the final DKH Hamiltonian, but the matrix multiplications to obtain the lower part result can be neglected. The symmetry of the matrices can also be exploited [647]. Noting that the odd matrices O are hermitean, = O, and the matrices Wjt are antihermitean, = —W/t- The algorithm requires the evaluation of their commutator. 

The resulting numbers of matrix multiplications required for the construction of the DKH Hamiltonian of orders 2 to 14 are listed in Table 14.1 [647]. 

Each element in a given row and column of the C matrix (the AB product) is obtained as follows. Select the corresponding row of the left-hand matrix A and the corresponding column of the right-hand matrix B. Multiply the first element of the row and the first element of the column together, then multiply the second element of the row and the second element of the column together, and so forth. The sum of all the products for this row and column selected will be the element of the AB product matrix C. The number of elements in the rows of the left-hand matrix must be the same as the number of elements in the column of the right-hand matrix in order for matrix multiplication to be possible. 

A group which consists of unitary matrices of fixed order with matrix multiplication as the group operation. 

The Gauss elimination procedure, which was described above in formula form, can also be accomplished by series of matrix multiplications. Two types of special matrices are involved in this operation. Both of these matrices are modifications of the identity matrix. The first type, which we designate as P,, is the identity matrix with the following changes The unity at position ii switches places with the zero at position ij, and the unity at position jj switches places with the zero at position ji. For example, for a fifth-order system is... 

To low orders, the expansion (10.7.37) is easily verified. For a given antisymmetric matrix X, we may then form the transformed density (10.7.32) by carrying out the expansion (10.7.37) to a sufficiently high order, thereby reducing the density transformation to a sequence of matrix multiplications and additions. Usually, only a few terms are used. However, since this truncation of the BCH expansion violates the trace and idempotency conditions, the resulting matrix must be purified as discussed in Section 10.7.6. 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

It is important to note that the product of two square matrices, given by AB is not necessarily equal to BA. In other words, matrix multiplication is not commutative. However, the trace of the product does not depend on the order of multiplication. From Eq. (28) it is apparent that... 

While the matrix multiplication defined by Eq. (28) is the more usual one in matrix algebra, there is another way of taking the product of two matrices. It is known as the direct product and is written here as A <8> 1 . If A is a square matrix of order n and B is a square matrix of order m, then A<8>B is a square matrix of order tun. Its elements consist of all possible pairs of elements, one each from A and B, viz. 

That is because of the way matrix multiplication is defined. Thus, for this case the order of appearance of the two matrices to be multiplied may provide different matrices as the answer. Thus, instead of f(x) and the expression for it in equation 1-1 describing the simple Normal distribution, the MND is described by the corresponding multivariate expression (1-2) ... 

---