Matrix algebra multiplication

Thus, the inverse matrix plays the role of division in matrix algebra. Multiplication of equation (1.33) from the left by B and from the right by C yields... 

The rules of matrix-vector multiplication show that the matrix form is the same as the algebraic form, Eq. (5-25)... 

While the matrix multiplication defined by Eq. (28) is the more usual one in matrix algebra, there is another way of taking the product of two matrices. It is known as the direct product and is written here as A <8> 1 . If A is a square matrix of order n and B is a square matrix of order m, then A<8>B is a square matrix of order tun. Its elements consist of all possible pairs of elements, one each from A and B, viz. 

Matrix Algebra and Multiple Linear Regression Part 1... 

In Chapters 2 and 3, we discussed the rules related to solving systems of linear equations using elementary algebraic manipulation, including simple matrix operations. The past chapters have described the inverse and transpose of a matrix in at least an introductory fashion. In this installment we would like to introduce the concepts of matrix algebra and their relationship to multiple linear regression (MLR). Let us start with the basic spectroscopic calibration relationship ... 

H. Mark, and J. Workman, Statistics in Spectroscopy Elementary Matrix Algebra and Multiple Linear Regression Conclusion , Spectroscopy 9(5), 22-23 (June, 1994). 

Basic understanding and efficient use of multivariate data analysis methods require some familiarity with matrix notation. The user of such methods, however, needs only elementary experience it is for instance not necessary to know computational details about matrix inversion or eigenvector calculation but the prerequisites and the meaning of such procedures should be evident. Important is a good understanding of matrix multiplication. A very short summary of basic matrix operations is presented in this section. Introductions to matrix algebra have been published elsewhere (Healy 2000 Manly 2000 Searle 2006). 

Whereas matrix addition (9.8) and scalar multiplication (9.9) have the usual associative and commutative properties of their scalar analogs, matrix multiplication (9.11), although associative [i.e., A(BC) = (AB)C], is inherently ncommutative [i.e., AB BAJ. This noncommutativity leads to some of the most characteristic and surprising features of matrix algebra, and underlies the still more surprising matrix-algebraic features of quantum theory. 

An algebra typically involves the operations of adding, subtracting, multiplying, or dividing the objects it describes, whether matrices or simple numbers. For completeness, we now summarize some other aspects of matrix algebra, built on the fundamental definitions of addition/subtraction (9.8), scalar multiplication (9.9), and matrix multiplication (9.11). 

Note that and A r need not be the same matrix (although they are for the important special case of real symmetric A that we are most concerned with). Note also that AT1 need not exist, even if A 0. A matrix for which A"1 exists is called nonsingular (see below) and leads to many arithmetic extensions that are not permitted to singular matrices. The many varieties of singularity (not just A = 0) and the (potentially) noncommutative aspects of multiplication distinguish matrix algebra from its scalar counterpart in interesting ways. 

It may be worth while to review the different kinds of multiplicity involved in the symbols appearing in Eqs. (3-6) and (3-15). Equation (3-6) is merely a shorthand way of writing the material balance for each of the key components, each term being a row matrix having as many elements as there are independent reactions. The equation asserts that when these matrices are combined as indicated, each element in the resulting matrix will be zero. The elements in the first two terms are obtained by vector differential operation, but the elements are scalars. Equation (3-15), on the other hand, is a scalar equation, from the point of view of both vector analysis and matrix algebra, although some of its terms involve vector operations and matrix products. No account need be taken of the interrelation of the vectors and matrices in these equations, but the order of vector differential operators and their operands as well as of all matrix products must be observed. 

The rule for the multiplication of determinants is so important that the reader is advised to consider the other proofs given in books on the subject, especially if he wishes to proceed to a study of matrix algebra,... 

Matrix algebra provides a powerful method for the manipulation of sets of numbers. Many mathematical operations — addition, subtraction, multiplication, division, etc. — have their counterparts in matrix algebra. Our discussion will be Umited to the manipulations of square matrices. For purposes of illustration, two 3x3 matrices will be defined, namely... 

Vector multiplication can be accomplished easily by the use of one of Excel s worksheet functions for matrix algebra, MMULT(inafr/x7, matrix2). For the matrices A and B defined above. 

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