Matrix operations in Matlab

Table 21-1 Matrix operations in MATLAB to compute equations 21-1-21-4...

This tutorial introduees the reader to matrix operations using MATLAB. All text in eourier font is either typed into, or printed into the eommand window. 

The few short programs presented in this chapter may also serve as a very rudimentary introduction into Matlab. Readers not familiar with Matlab but otherwise proficient in an alternative language will be surprised at the almost complete lack of for. .. end loops. We also introduce matrix operations in Excel, assuming that the other, more common aspects of Excel are known to the reader. While there is a reasonable collection of matrix operations available in Excel, their usage is rather cumbersome. We believe that many readers will appreciate the short introduction into this aspect of Excel. 

In contrast to Matlab, where the defaults are the matrix operators, in Excel the default is the element-wise operation. In fact, all basic operations (e-g- 0, 0> 0> 0> Q) and functions (e.g. EXP, LN, LOG) work element-wise in Excel. All... 

MATLAB offers many tools for matrix operations and manipulations. Table 15.5 shows examples of these capabilities. We will explain MATLAB s matrix operations in more detail in Section 15.5. 

The operation is the left division and / is right division. In this case, left division is used since the matrix A is to the left of vector b. You can learn about these operations in MATLAB by entering help slash. The right division B/A computes (A /B ) again using LU factorization and Guassian eliminination. 

In the second part of the function, locations of all the points at which the values of the function are to be evaluated are found in between the base points. Because matrix operations are much faster than element-by-element operations in MATLAB, the required number of independent and dependent variables are arranged in two interim matrices at each location. These matrices are used at the interpolation section for doing the interpolation in vector form. 

Now using the MATLAB command line software, we can easily demonstrate this solution (for the multivariate problem we have identified) using a series of simple matrix operations as shown in Table 21-1 below ... 

For the next several chapters in this book we will illustrate the straight forward calculations used for multivariate regression. In each case we continue to perform all mathematical operations using MATLAB software [1, 2], We have already discussed and shown the manual methods for calculating most of the matrix algebra used here in references [3-6]. You may wish to program these operations yourselves or use other software to routinely make these calculations. 

Matlab is a matrix oriented language that is just about perfect for most data analysis tasks. Those readers who already know Matlab will agree with that statement. Those who have not used Matlab so far, will be amazed by the ease with which rather sophisticated programs can be developed. This strength of Matlab is a weak point in Excel. While Excel does include matrix operations, they are clumsy and probably for this reason, not well known and used. An additional shortcoming of Excel is the lack of functions for Factor Analysis or the Singular Value Decomposition. Nevertheless, Excel is very powerful and allows the analysis of fairly complex data. 

The capability of dragging results from one cell to others is a very useful property of Excel and becomes even more powerful in combination with the dollar operator ( ) correctly applied within the cell reference. Referring to the previous Matlab example, if the scalar element foi,2 (cell F3) is to be subtracted from matrix A (A3 C4) in Excel, putting the dollar operator ( ) in front of the column and row reference of the source cell containing the scalar foi,2 ( F 3), prevents "dragging-over" of the source cell F3 in both column and row direction. 

In Matlab the asterisk operator ( ) is used for the matrix product. If the corresponding dimensions match all individual scalar products, c xarj, are evaluated to form Y. 

As stated earlier, Matlab s philosophy is to read everything as a matrix. Consequently, the basic operators for multiplication, right division, left division, power (, /,, A) automatically perform corresponding matrix operations (A will be introduced shortly in the context of square matrices, / and will be discussed later, in the context of linear regression and the calculation of a pseudo inverse, see The Pseudo-Inverse, p.117). 

Finally, many chemometricians use matrices. Matlab (see Section A.5) is better than Excel for developing sophisticated matrix based algorithms, but for simple applications the matrix operations of Excel can also be translated into VBA. As discussed in Section A.4.4.2, this is somewhat awkward, but for the specialist programmer there is a simple trick, which is to break down the matrix expression into a character string. Strings can be concatenated using die sign. Therefore, the expression... 

A significant advantage of Matlab is that there are several further veiy useful matrix operations. Most are in the form of functions the arguments are enclosed in brackets. Three that are important in chemometrics are as follows ... 

These matrix operations are easily programmed with mathematical software packages. In MATLAB, for example, only two statements, M = A A and [V,S] = eig(M) , are required... 

The transpose of a matrix M is calculated using the command M. The inverse of a matrix M is calculated as inv(M). In MATLAB, the multiphcation of matrices A and B is denoted by A B, while their addition and subtraction are denoted by A -h B and A - B, respectively. Commands for element-by-element multiphcation and division are also available. For more functions and help on any MATLAB operation, type help. 

A (-l). The expression AIB in MATLAB is equivalent to AB and the expression A B is equivalent to A B. Note that the expression A./B (putting before division operator) is element-by-element division of the elements of the two matrices and the expression A A(-l) results in a matrix whose elements are the reciprocals of the elements of the original matrix. 

MATLAB is designed to make operations on matrices as easy as possible. Most of the variables in MATLAB are considered as matrices. A scalar number is a 1x1 matrix and a vector is a Ixn (or nxl) matrix. Introducing a matrix is also done by an equality sign ... 

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