Matrix Computation with Excel

matrix [AT] is a three by three (or 3X3) matrix whose elements are numbers, and L] is a three by one matrix with its elements representing variables x, y, and ss. The [AH is called a square matrix. A sqmre matrix has the same niunber of rows and columns. The element of a matrix is denoted by its location. For example, the element in the first row and the third column of matrix [AT] is denoted 1 i3, which has a value of nine. In this book, we denote the matrix by a boldfiice letter in brackets [] and, for example [AI], [T], F, and the elements of matrices are represented by regular lower case letters. The are used to distinguish a column matrix. A column matrix is defined as a matrix that has one column but could have many rows. On the other hand, a row matrix is a matrix that has one row but could have many columns. 

In Section 18.5, we will discuss matrix algebra in mote detail. If you do not have an adequate background in matrix algebra, you may want to read Section 18.5 before studying the following examples. 

Usii Example 14.8, we will show how to use Excel to peifbnn certain matrix operations. 

If you do not have any background in matrix a bra, you may want to study Section 18.5 to learn about matrix operation rules. The manual hand calculations for this example problem is also shown in that section. 

Refer to the Excel sheets shown in the accompanying figures when following the steps. 

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In Figure 19-5, we enter the wavelengths in column A just to keep track of information. We will not use these wavelengths for computation. Enter the products eh for pure X in column B and eh for pure Y in column C. The array in cells B5 C6 is the matrix K. The Excel function MINVERSE(B5 C6) gives the inverse matrix, K-1. The function MMULT(matrix 1, matrix 2) gives the product of two matrices (or a matrix and a vector). The concentration vector, C, is equal to K 1 A, which we get with the single statement... 

The relative simplicity of density functional theory (DFT) calculations involving H has made this area extremely well served by computational studies. Excellent reviews are available. " Oxidative addition of H2 can proceed with low or zero barrier for the most exothermic reactions. In contrast to Pt + H2 that gave a low barrier oxidative addition of this sort, the Pd + H2 system was shown to have a stable Pd(H2) in matrix in 1986. ... 

A numerical solution of the Schrodinger equation in Eq. [1] often starts with the discretization of the wave function. Discretization is necessary because it converts the differential equation to a matrix form, which can then be readily handled by a digital computer. This process is typically done using a set of basis functions in a chosen coordinate system. As discussed extensively in the literature,5,9-11 the proper choice of the coordinate system and the basis functions is vital in minimizing the size of the problem and in providing a physically relevant interpretation of the solution. However, this important topic is out of the scope of this review and we will only discuss some related issues in the context of recursive diagonalization. Interested readers are referred to other excellent reviews on this topic.5,9,10... 

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