Algebra matrices

Matrix algebra provides a concise and practical method for carrying out the mathematical operations involved in the design of experiments and in the treatment of the resulting experimental data. 

A matrix is a rectangular array of numbers. Many types of data are tabulated in arrays. For example, baseball fans are familiar with a tabulation of data similar to the following array  

Not only is the value of each element in the matrix important, but the location of each element is also significant. Fans of the Atlanta team would be dismayed to see the sixth row of the array, 

If we omit the row and column headings and focus our attention on the arrays of numbers in this example, we are dealing with the matrices 

The dimensions of a matrix are given by stating first the number of rows and then the number of columns that it has. Thus, matrix A has six rows and three columns, and is said to be a 6x3 (read six by three ) matrix. Matrix B has one row and three columns and is a 1x3 matrix. Matrix C is a 6x1 matrix. Generally, a matrix M that has r rows and c columns is called an rxc matrix and can be identified as such by the notation 

a square matrix representing the second-rank tensor C Then the scalar product of vectors can be written as aTb = bTa = aaba = d 

Some matrix rules for square matrices are collected in Table 1.2. 

The tridiagonal matrix is encountered quite regularly when solving differential equations using the finite-diflference method (see Chapter 12). 

The transpose of a IV X Af matrix A is a matrix having a size of M x N, with the element af. defined as 

A symmetric square matrix has identical elements on either side of the major diagonal line, that is, ay,- = a,-y. This means A = A. 

A sparse matrix is a matrix in which most elements are zero. Many matrices encountered in solving engineering systems are sparse matrices. 

A diagonally dominant matrix is a matrix such that the absolute value of the diagonal term is larger than the sum of the absolute values of other elements in the same row, with the diagonal term larger than the corresponding sum for at least one row that is. 

Thus far, we have been doing algebra involving numbers and functions. It is also possible to apply the operations of algebra to more general types of mathematical entities. In this chapter, we will deal with matrices, which are ordered arrays of numbers or functions. For exanple, a matrix that we designate by tbe symbol A can represent a collection of quantities arrayed as follows  

The subscripts i and j on the matrix elements aij label the rows and columns, respectively. The matrix A shown above is an n x n square matrix, with n rows and n columns. We will also make use of n x 1 column matrices or column vectors such as 

Prove that tr(AB) = tr(BA) where A and B are any two matrices that are conformable for both multiplications. They need not be square. 

Prove for that for Kx 1 column vectors, x, 1. , and some nonzero vector, a, 

Since ( x - a) is a vector of constants, it may be moved out of the summations. Thus, the fourth tenn is 

B is called a permutation matrix. Each column of B, say, b is a column of an identity matrix. The /th column of the matrix product AB is A b, which is the /th column of A. Therefore, post multiplication of A by B simply rearranges (permutes) the columns of A (hence the name). Each row of the product BA is one of the rows of A, so the product BA is a rearrangement of the rows of A. Of course, A need not be square for us to permute its rows or columns. If not, the applicable pennutation matrix will be of different orders for the rows and columns. 

Since each power of B is a rearrangement of I, some power of B will equal I. If n is this power, we also find, therefore, that B 1 = B. This will hold generally. 

The rth diagonal element of AB is Z.jOijbji Summing over i produces tr(AB) = T.iHi ijbji. The jth diagonal element of BA is T.jbj a j. Summing over / produces tr(BA) = T.iHjbjjajj.  

Finally, the followii are some addidonal mathemadcal reladonships that you may find helpfrd during your engineering educadon. 

During your engineering educadon, you will leam about diflktent types of physical variables. There are those that are identifiable by a single value or magnitude. For example, time can be described by a single wdue such as two hours. These r pes of physical variables which are 

Diagonal and Unit Matrices A diagonal matrix is one that only has elements along its principal diagonal the elements are zero everywhere else. An example of a 4 X 4 diagonal matrix firllows. 

The diagonal along which values 5,7j 4, and 11 lies is called the principal diagonal An identity or unit matrix is a diagonal matrix whose elements consist of a value of 1. An example of an identity matrix follows. 

Two matrices can be added nether or subtraaed from each other provided that they are of the same size—each matrix must have the same number of rows and columns. We can add matrix [A] Xtt of dimension mhy n (having m rows and n columns) to matrix of dte 

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Central the molecular graph is completely coded (each atom and bond is represented) matrix algebra can be used the niimber of entries in the matrix grows with the square of the number of atoms in ) no stereochemistry included... 

The unit matrix plays the same role in matrix algebra that 1 plays in ordinar y algebra. Multiplieation of a matr ix by the unit matrix leaves it unehanged ... 

The unit matrix, I, with an = 1 and Gy = 0 for i plays the same role in matrix algebra that the number 1 plays in ordinary algebra. In ordinary algebra, we can perform an operation on any number, say 5, to reduce it to 1 (divide by 5). If we do the same operation on 1, we obtain the inverse of 5, namely, 1/5. Analogously, in matrix algebra, if we cany out a series of operations on A to reduce it to the unit matrix and cany out the same series of operations on the unit matrix itself, we obtain the inverse of the original matrix A . ... 

In multivariate least squares analysis, the dependent variable is a function of two or more independent variables. Because matrices are so conveniently handled by computer and because the mathematical formalism is simpler, multivariate analysis will be developed as a topic in matrix algebra rather than conventional algebra. 

This book is an introduction to computational chemistr y, molecular mechanics, and molecular orbital calculations, using a personal mieroeomputer. No speeial eom-putational skills are assumed of the reader aside from the ability to read and write a simple program in BASIC. No mathematieal training beyond ealeulus is assumed. A few elements of matrix algebra are introdueed in Chapter 3 and used throughout. 

Rules of matrix algebra can be appHed to the manipulation and interpretation of data in this type of matrix format. One of the most basic operations that can be performed is to plot the samples in variable-by-variable plots. When the number of variables is as small as two then it is a simple and familiar matter to constmct and analyze the plot. But if the number of variables exceeds two or three, it is obviously impractical to try to interpret the data using simple bivariate plots. Pattern recognition provides computer tools far superior to bivariate plots for understanding the data stmcture in the //-dimensional vector space. 

The example demonstrates that not all the B-numbers of equation 5 are linearly independent. A set of linearly independent B-numbers is said to be complete if every B-number of Dis a product of powers of the B-numbers of the set. To determine the number of elements in a complete set of B-numbers, it is only necessary to determine the number of linearly independent solutions of equation 13. The solution to the latter is well known and can be found in any text on matrix algebra (see, for example, (39) and (40)). Thus the following theorems can be stated. 

E. E. Hohn, Elementaj Matrix Algebra., The Macmillan Co., New York, 1958. 

The solution of the system may then be found by elimination or matrix methods if a solution exists (see Matrix Algebra and Matrix Computations ). 

See the section entitled Matrix Algebra and Matrix Computation. ... 

Matrix algebra also involves the addition and subtraction of matrices. The rules for this are as follows ... 

Because the stiffness and compliance matrices are mutually inverse, it follows by matrix algebra that their components are related as follows for orthotropic materials ... 

The summations in Eqs. (2-73) are over all i. Equations (2-73) are called the normal regression equations. With the experimental observations of 3, as a function of the Xij, the summations are carried out, and the resulting simultaneous equations are solved for the parameters. This is usually done by matrix algebra. Define these matrices ... 

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