Example matrix inversion

A linear coordinate transformation may be illustrated by a simple two-dimensional example. The new coordinate system is defined in term of the old by means of a rotation matrix, U. In the general case the U matrix is unitary (complex elements), although for most applications it may be chosen to be orthogonal (real elements). This means that the matrix inverse is given by transposing the complex conjugate, or in the... 

In principle, the relationships described by equations 66-9 (a-c) could be used directly to construct a function that relates test results to sample concentrations. In practice, there are some important considerations that must be taken into account. The major consideration is the possibility of correlation between the various powers of X. We find, for example, that the correlation coefficient of the integers from 1 to 10 with their squares is 0.974 - a rather high value. Arden describes this mathematically and shows how the determinant of the matrix formed by equations 66-9 (a-c) becomes smaller and smaller as the number of terms included in equation 66-4 increases, due to correlation between the various powers of X. Arden is concerned with computational issues, and his concern is that the determinant will become so small that operations such as matrix inversion will be come impossible to perform because of truncation error in the computer used. Our concerns are not so severe as we shall see, we are not likely to run into such drastic problems. 

Another use for the matrix inverse is to express one set of variables in terms of another, an important operation in constrained optimization (see Chapter 8). For example, suppose x and z are two n- vectors that are related by... 

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

In Excel, matrix inversion can be performed similarly to matrix transposition (see earlier). Figure 2-13 gives an example. Cells D3 E4, defining the target matrix, have to be pre-selected and now the MINVERSE function is applied to the source cells A3 B4. Finally, the SHIFT+CTRL+ENTER key combination is used to confirm the matrix operation. 

If the determinant of the matrix to be inverted is zero, the calculations to be performed are undefined. This suggests a general rule a square matrix has an inverse if and only if its determinant is not equal to zero. A matrix having a zero determinant is said to be singular and has no inverse. As an example of matrix inversion, consider the 2x2 matrix... 

In Example 1.1.4 we could replace e by the vector ai in the basis for all i. Matrix inversion (or solution of a matrix equation) is, however, not always as simple. Indeed, we run into trouble if we want to replace by... 

Real problems are likely to be considerably more complex than the examples that have appeared in the literature. It is for this reason that the computer assumes a particular importance in this work. The method of solution for linear-programming problems is very similar, in terms of its elemental steps, to the operations required in matrix inversions. A description ot the calculations required for the Simplex method of solution is given in Charnes, Cooper, and Henderson s introductory book on linear programming (C2). Unless the problem has special character-... 

All of the matrices we have just worked out, as well as all others which describe the transformations of a set of orthogonal coordinates by proper and improper rotations, are called orthogonal matrices. They have the convenient property that their inverses are obtained merely by transposing rows and columns. Thus, for example, the inverse of the matrix... 

Note that the second integral only plays a role if the point G [0,1/2], We now have a linear system of equations that will require a conventional matrix inversion method. If, for example N = 2, the system becomes... 

One possible way to solve this problem is by obtaining a system of linear equations in a similar way as in Example 8.3. However, the equations must be organized carefully because the equations are coupled in the two directions. To avoid the complications given by the generation of the matrix for the linear system of equations, some iterative methodologies have been developed to solve for this type of problems. Such techniques solve the system of equations without the need of cumbersome matrix manipulation, such as LU-decomposition, matrix inversion, etc. [19]. 

In some programs such as CVRUCAT or the subroutine U DERIV, matrix inversion is needed. This is best done by using LU decomposition, as described in Press et al. (1986). The subroutine MATINV does this. It assumes a square matrix, of the exact size given, so the best way to call it is by using a section of that size, for example... 

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

Spreadsheet Summary In some chemical problems, two or more 3 simultaneous equations must be solved to obtain the desired result. Example 12-3 is such a problem. In Chapter 6 of Applications of Microsoft Excel in Analytical Chemistry, the method of determinants and the matrix inversion method are explored for solving such equations. The matrix method is extended to solve a system of 4 equations in 4 unknowns. The matrix method is used to confirm the results of Example 12-3. 

Finally, for this section we note that the valence interactions in Eq. [1] are either linear with respect to the force constants or can be made linear. For example, the harmonic approximation for the bond stretch, 0.5 (b - boV, is linear with respect to the force constant If a Morse function is chosen, then it is possible to linearize it by a Taylor expansion, etc. Even the dependence on the reference value bg can be transformed such that the force field has a linear term k, b - bo), where bo is predetermined and fixed, and is the parameter to be determined. The dependence of the energy function on the latter is linear. [After ko has been determined the bilinear form in b - bo) can be rearranged such that bo is modified and the term linear in b - bo) disappears.] Consequently, the fit of the force constants to the ab initio data can be transformed into a linear least-squares problem with respect to these parameters, and such a problem can be solved with one matrix inversion. This is to be distinguished from parameter optimizations with respect to experimental data such as frequencies that are, of course, complicated functions of the whole set of force constants and the molecular geometry. The linearity of the least-squares problem with respect to the ab initio data is a reflection of the point discussed in the previous section, which noted that the ab initio data are related to the functional form of empirical force fields more directly than the experimental data. A related advantage in this respect is that, when fitting the ab initio Hessian matrix and determining in this way the molecular normal modes and frequencies, one does not compare anharmonic and harmonic frequencies, as is usually done with respect to experimental results. 

As you have seen, matrix algebra can be tedious. Mathematica has all of the matrix operations built into it, so that you can form matrix products and carry out matrix inversion automatically. Mathematica treats matrices as lists of lists, with the elements of each row entered as a list. A list is entered inside curly brackets ( braces ) with the elements separated by commas. A list of lists requires braces around the set of lists with braces and commas. For example, to enter the following 3 by 3 matrix... 

Equation (2) also can be solved by other methods without direct implementation of matrix inverse transformation K for example, by means of linear iterations ... 

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