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By Anubhav Diwan and Matthew R. Linford, Contributing Editor A Brief Introduction to Matrix Algebra W e will cover the basics of matrix algebra here. We will (i) describe what a matrix is, and then discuss (ii) matrix addition, (iii) matrix multiplication by a con-stant, (iv) multiplication of matrices, (v) the identity matrix, (vi) the inverse of a matrix, (vii) an algorithm for ﬁnding the inverse of a matrix, and (viii) the transpose of a matrix. We will then use matrix algebra to (ix) solve the simple problem of ﬁtting a straight line between two points, and (x) the slightly more com-plex problem of ﬁtting a straight line to three points. Finally, we will discuss (xi) eigenvectors and eigenvalues, including (xii) an algorithm to ﬁnd them. While on the topic of matrix algebra we will describe three Excel commands you should probably know if you are going to manipulate matrices, and show a nice trick for exporting data from Excel into MATLAB. Excel and MATLAB are powerful tools for manipulating numerical data. To test your understanding of matrix algebra, a few problems (with solutions) are then presented. Introduction Chemometrics is the branch of analytical chemistry that deals with data analysis. It is really just the statistical analysis of data, and it includes all sorts of tools for smoothing data, removing baselines, peak recognition, peak ﬁtting, recognizing patterns/ categorizing data, etc. Different disciplines have different names for these tools. When the same statistical methods are applied to biological data, the ﬁeld is called bioinformatics. Whatever we want to call these methods, perhaps just the ‘statistical analysis of data’, they are often very important in surface and material analysis – they can be important for getting the most information possible out of one’s data. Matrix algebra is very widely used for chemometrics/bioinformatics/the statistical analysis of data and is the topic of this contribution. Some of the Basics of Matrix Algebra This quick overview of matrix algebra starts at a pretty basic level. If it overwhelms you, you will probably want to consult an elementary textbook on the subject. 1. A matrix is an array of objects, e.g., numbers or vari-ables, with a certain number of rows and columns that is typically placed between square brackets. For exam-ple, the following matrix has two rows and ﬁve columns: D = . We describe it as a 2 × 5 matrix, i.e., when we write the dimensions of a matrix we list the number of rows followed by the number of columns. The individual objects in a matrix can be addressed by giving the row number followed by the column number as follows: D 11 = 1, D 13 = a , and D 25 = r . 2. Matrices can be added only if their dimensions are identi-cal. That is, an ( s × r ) matrix can be added to a ( t × v ) matrix only if s = t and r = v . Matrices are added element by element. Example 1. Addition of two matrices. 3. If a matrix is multiplied by a constant every element in the matrix is multiplied by that constant. Example 2 . Multiplication of a matrix by a constant. 6ACUUM4ECHNOLOGY#OATINGs January 2015 www.vactechmag.com or www.vtcmag.com WP1

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