Properties of Matrices

An interesting and complicated problem is to characterize the properties of matrices that are actually incidence matrices of graphs. This question has been resolved and leads to several interesting ideas. 

One of the air of multivariate analysis is to reveal patterns in the data, whether they are in the form of a measurement table or in that of a contingency table. In this chapter we will refer to both of them by the more algebraic term matrix . In what follows we describe the basic properties of matrices and of operations that can be applied to them. In many cases we will not provide proofs of the theorems that underlie these properties, as these proofs can be found in textbooks on matrix algebra (e.g. Gantmacher [2]). The algebraic part of this section is also treated more extensively in textbooks on multivariate analysis (e.g. Dillon and Goldstein [1], Giri [3], Cliff [4], Harris [5], Chatfield and Collins [6], Srivastana and Carter [7], Anderson [8]). 

An explicit set of operators Oy, Oy, Oz with the foregoing properties can be formed using 2X2 matrices. The properties of matrices are discussed in Appendix I. In matrix notation, equation (7.19) is... 

The bold-face characters employed in Eq. (33) imply that each symbol represents a matrix. The problem of the resolution of simultaneous linear equations will be discussed in Section 7.8, as certain properties of matrices must first be explained. 

An important concept is that of spectral properties of matrices. A matrix can be diagonalized if there is a basis in which the matrix is diagonal. If so, there exists a basis set transformation, in this context called similarity transformation, of the form... 

Binary composition in a set of abstract elements g,, whatever its nature, is always written as a multiplication and is usually referred to as multiplication whatever it actually may be. For example, if g, and g, are operators then the product g,- gy means carry out the operation implied by gy and then that implied by g,. If g, and gy are both -dimensional square matrices then g, gy is the matrix product of the two matrices g, and gy evaluated using the usual row x column law of matrix multiplication. (The properties of matrices that are made use of in this book are reviewed in Appendix Al.) Binary composition is unique but is not necessarily commutative g, g, may or may not be equal to gy gt. In order for a set of abstract elements g, to be a G, the law of binary composition must be defined and the set must possess the following four properties. 

This is the anticipated result since the MRs of symmetry operators obey the same multiplication table as the operators themselves, and it is known from the properties of matrices that... 

In another study, fluorescent spectroscopy was used to compare the physicochemical properties of matrices constructed either from the gel-state PC or from the stratum corneum lipids. The transition temperatures were found to be 55°C for the gel-state PCs and 60 to 63°C for the stratum corneum lipids. Further comparison showed that the gel-state PC is more hydrophilic and therefore binds more water. It is also more fluid and more polar.34... 

Important information about the symmetry aspects of point groups is summarized in character tables, described later in this chapter. To understand the construction and use of character tables, we first need to consider the properties of matrices, which are the basis for the tables. ... 

A noteworthy property of matrices found in Eqs. 1.21, 1.22 and 1.24 to 1.26 is their unimodularity - the determinant of every matrix is equal to 1 or -1 for the rotation and inversion (or roto-inversion) operations, respectively, which is shown for the rotation around Z in Eq. 1.27. 

This makes evident the projector properties of matrices Pi and P2. Furthermore, matrices Pi and P2 allow one to write matrix A in the so-called canonical form ... 

Several generic optical characteristics of multi-layer specimens can be derived with reference to the mathematical properties of matrices (101. 

Step 2 iSliift the packing cell C leftward and downward until it meets other packing cells and cannot be moved again. In view of the property of matrices, it is convenient to shift the packing cell by counting the empty cells in the matrix. Then the matrix of the stock sheet becomes... 

The algebraic properties of matrices can be found in the literature on linear algebra [2-5]. Here we list those that facilitate the discussions in Chapter 9. Proofs are left to the linear algebra literature ... 

For more complete discussions of the properties of matrices, sec the following ... 

Cook KD, Todd PJ, Friar DPI. Physical properties of matrices used for fast atom bombardment. Biomed Environ Mass Spectrom. 1989 18 492-7. 

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