Matrices multiplications

The matrix equation [E.l] involves the multiplication of the matrices [A] and (x). To do this one must apply the simple rules of matrix multiplication. These are  

The number of columns N in the first matrix must be equal to the number of rows in the second matrix. Nonsquare matrices can be multiplied. The order of multiplication is important, that is, 48 i not equal to 84- Matrix multiplication does nor commute as the mathematicians say. There is a difference between premultiplying and postmultiplying a matrix by another matrix. 

Example 15.2. Find the product of the and g matrices given in Example 15.1 (u) when d is multiplied by and (I ) when g is multiplied by 

This example illustrates that matrix multiplication does not commute. 

The transpose of a matrix is another matrix which has columns that are the same as the rows of the original matrix. 

If the matrix 4 is defined as the transpose of the matrix 4, each element in is given by 

The product of two matrices AB exists if and only if the number of rows in the second matrix B is the same as the number of columns in the first matrix A. If this is the case, the two matrices are said to be conformable for multiplication. If A is an mxp matrix and B is a pxn matrix, then the product C is an mxn matrix  

Each of the elements of the product matrix, C = AB, is found by multiplying each of the p elements in a column of B by the corresponding p elements in a row of A and taking the sum of the intermediate products. Algebraically, an element c,y is calculated 

An alternative layout of the matrices is often useful, especially when the matrices are large. The right matrix is raised, and the product matrix is moved to the left into the space that has been made available. The rows of the left matrix and the columns of the right matrix now point to the location of the corresponding product element. 

Another example of matrix multiplication involves an identity matrix  

Where do matrices come from Suppose we have a set of n simultaneous relations, each involving n quantities xi, X2, X3. .. x  

This set of n relations can be represented symbolically by a single matrix equation 

Comparing Eq. (9.4) with Eq. (9.5), it is seen that matrix multiplication implies the following rule involving their component elements  

Note that summation over identical adjacent indices k results in their mutual annihilation. Suppose the quantities yi in Eq. (9.4) are themselves determined by n simultaneous relations 

An element of the product matrix is constructed by summation over two sets of matrix elements in the following pattern  

Any two conformable matrices A and B can be multiplied in the order AB. A and B are conformable if the number of columns of A is the same as the number of rows of B. This is summarized as follows, where Aisnxp and Bispxm  

Another view is that the t.j-th element of C is the inner product of the ith row of A with the yth column of B. 

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By perfomiing the matrix multiplication, one can get the following relations between the adiabatic eigenvalues j(0) and the D matiix elements... 

By applying Eq. (C.13) to the spin operators Si and using Eq. (C.22), one then gets after some matrix multiplications... 

US model can be combined with the Monte Carlo simulation approach to calculate a r range of properties them is available from the simple matrix multiplication method. 2 RIS Monte Carlo method the statistical weight matrices are used to generate chain irmadons with a probability distribution that is implied in their statistical weights. 

If a vector x is transformed into a new vector x by a matrix multiplication... 

The left side of the normal equations can be seen to be a product including X, its transpose, and m. Matrix multiplication shows that... 

Having filled in all the elements of the F matr ix, we use an iterative diagonaliza-tion procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a macroiteration. The term is descriptive and we shall use it from time to time. 

As another example, consider the following matrix multiplication ... 

Matrix multiplication happens to be cummutative in this special case. It is easy to raise a matrix to a power on a computer since three nmlti-plications give the eighth power, etc. Therefore the matrix formulation is well adapted to computer use. 

Equations la and lb are for a simple two-phase system such as the air-bulk solid interface. Real materials aren t so simple. They have natural oxides and surface roughness, and consist of deposited or grown multilayered structures in many cases. In these cases each layer and interface can be represented by a 2 x 2 matrix (for isotropic materials), and the overall reflection properties can be calculated by matrix multiplication. The resulting algebraic equations are too complex to invert, and a major consequence is that regression analysis must be used to determine the system s physical parameters. ... 

For a more complicated [B] matrix that has, say, n columns whereas [A] has m rows (remember [A] must have p columns and [B] must have p rows), the [C] matrix will have m rows and n columns. That is, the multiplication in Equations (A.21) and (A.22) is repeated as many times as there are columns in [B]. Note that, although the product [A][B] can be found as in Equation (A.21), the product [B][A] is not simultaneously defined unless [B] and [A] have the same number of rows and columns. Thus, [A] cannot be premultiplied by [B] if [A][B] is defined unless [B] and [A] are square. Moreover, even if both [A][B] and [B][A] are defined, there is no guarantee that [A][B] = [B][A]. That is, matrix multiplication is not necessarily commutative. 

The SSW form an ideal expansion set as their shape is determined by the crystal structure. Hence only a few are required. This expansion can be formulated in both real and reciprocal space, which should make the method applicable to non periodic systems. When formulated in real space all the matrix multiplications and inversions become 0(N). This makes the method comparably fast for cells large than the localisation length of the SSW. In addition once the expansion is made, Poisson s equation can be solved exactly, and the integrals over the intersitital region can be calculated exactly. 

We can also use the definition of matrix multiplication to write equation [21 ] as a matrix equation ... 

Again, please consult Appendix A if you are not yet comfortable with matrix multiplication. 

Taking advantage of the associative property of matrix multiplication, we can compute the quantity [KTK] KT at calibration time. 

Thus, we can predict the concentrations in an unknown by a simple matrix multiplication of a calibration matrix and the unknown spectrum. 

From the definition of matrix multiplication, we see that the product of any matrix multiplied with a properly dimensioned zero matrix must be a zero matrix. We also see that the any matrix that is multiplied with a properly dimensioned unit matrix will remain unchanged by the multiplication. 

The operation of matrix multiplication can be shown to be associative, meaning that X(YZ) = (XY)Z. But, it is not commutative, as in general we will have that XY YX. Matrix multiplication is distributive with respect to matrix addition, which implies that (X + Y)Z = XZ + YZ. When this expression is read from right to left, the process is called factoring-out [4]. 

Matrix multiplication can be applied to vectors, if the latter are regarded as one-column matrices. This way, we can distinguish between four types of special matrix products, which are explained below and which are represented schematically in Fig. 29.6. 

The PLS algorithm is relatively fast because it only involves simple matrix multiplications. Eigenvalue/eigenvector analysis or matrix inversions are not needed. The determination of how many factors to take is a major decision. Just as for the other methods the right number of components can be determined by assessing the predictive ability of models of increasing dimensionality. This is more fully discussed in Section 36.5 on validation. 

After completing the matrix multiplication operations, we obtain... 

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.

The solution of Equation 10.28 is obtained in one step by performing a simple matrix multiplication since the inverse of the matrix on the left hand side of Equation 10.28 is already available from the integration of the state equations. Equation 10.28 is solved for r=l,...,p and thus the whole sensitivity matrix G(tr,) is obtained as [gi(tHt), g2(t,+1),- - , gP(t,+i)]. The computational savings that are realized by the above procedure are substantial, especially when the number of unknown parameters is large (Tan and Kalogerakis, 1991). With this modification the computational requirements of the Gauss-Newton method for PDE models become reasonable and hence, the estimation method becomes implementable. 

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. 

The matrix represented in this chapter by A is usually called the adjoint matrix. It is obtained by constructing the matrix which is composed of all of the cofactors of the elements a,j in A and then taking its transpose. With the basic definition of matrix multiplication (Eq. (29)J and some patience, die reader can verify the relation... 

This result should be obvious from the definition of matrix multiplication fEq. (28)]. 

Cany out the matrix multiplication indicated in Eq. (17) to verily Table 1. 

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