Product of two matrices

State-of-the-art for data evaluation of complex depth profile is the use of factor analysis. The acquired data can be compiled in a two-dimensional data matrix in a manner that the n intensity values N(E) or, in the derivative mode dN( )/d , respectively, of a spectrum recorded in the ith of a total of m sputter cycles are written in the ith column of the data matrix D. For the purpose of factor analysis, it now becomes necessary that the (n X m)-dimensional data matrix D can be expressed as a product of two matrices, i. e. the (n x k)-dimensional spectrum matrix R and the (k x m)-dimensional concentration matrix C, in which R in k columns contains the spectra of k components, and C in k rows contains the concentrations of the respective m sputter cycles, i. e. ... 

From a previously derived property of the trace of a product of two matrices (eq. (29.26)) follows that ... 

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. 

The product of two matrices is therefore similar to the scalar product of two vectors. C is the product of AB, according to... 

The direct product of two matrices is best explained in terms of an example. 

The Jacobian matrix of any metabolic network can be written as product of two matrices [23, 84, 325]. Consider the metabolic balance equation, describing the time-dependent behavior of the concentration. S, -(7),... 

Other notation used diagB is the diagonal n x n matrix consisting of the diagonal elements of the square matrix B. The trace of B is denoted trB, and the determinant of B is denoted B. The Kronecker product of two matrices is denoted by symbol (g). Other notation will be introduced as needed. 

The product of two matrices AB exists if and only if the number of rows in the second matrix B is the same as the number of columns in the first matrix A. If this is the case, the two matrices are said to be conformable for multiplication. If A is an mxp matrix and B is a pxn matrix, then the product C is an mxn matrix ... 

From consideration of the formula for the product of two matrices, it is apparent that the above relationship leads to... 

The direct product of two matrices is quite different from the ordinary matrix product. First, let us consider how the indices of the various matrix elements are related, By comparing eqns (8-3.2) and (8-3.4) we have f, = ... 

In Figure 19-5, we enter the wavelengths in column A just to keep track of information. We will not use these wavelengths for computation. Enter the products eh for pure X in column B and eh for pure Y in column C. The array in cells B5 C6 is the matrix K. The Excel function MINVERSE(B5 C6) gives the inverse matrix, K-1. The function MMULT(matrix 1, matrix 2) gives the product of two matrices (or a matrix and a vector). The concentration vector, C, is equal to K 1 A, which we get with the single statement... 

If one reads the integral as a product of two matrices, or integral kernels, this equation may be written... 

Let us denote by the symbol Am n a matrix with m rows and n columns. The symbols A1,, A" , and A1,n then denote a scalar, a column vector, and a row vector, respectively. The direct product of two matrices is defined by equation (A4) (see, for example, reference 46) ... 

The loading and scores for PCA can be generated by singular value decomposition (SVD). Instead of expressing the matrix containing the mixture spectra, A, as a product of two matrices as in Equation (4.4), SVD expresses it as a product of three matrices... 

Null space. If the product of two matrices is a zero matrix (all zeros), ax = 0 is said to be a homogeneous equation. The matrix jc is said to be the null space of a. Tn Mathematica a basis for the null space of a can be calculated by use of Null Space [a]. There is a degree of arbitrariness in the null space in that it provides a basis, and alternative forms can be calculated from it, that are equivalent. See Equation 5.1-19 for a method to calculate a basis for the null space by hand. When a basis for the null space of a matrix needs to be compared with another matrix of the same dimensions, they are both row reduced. If the two matrices have the same row-reduced form, they are equivalent. 

Y can be written as a product of two matrices C and A, where C contains as columns the concentration profiles of the absorbing species. If there are nc absorbing species, C has nc columns, each one containing nt elements, the concentrations of the species at the nt reaction times. Similarly, the matrix A contains, in nc rows, the molar... 

In a more formal sense, the original Raman spectra of a set of mixtures containing various concentrations of the desired components can be set up in a matrix format, with each row of the matrix containing the intensities of each Raman spectrum. This matrix, which we will call R, contains m rows of spectra, each with w frequencies. PCA expresses this R matrix as a product of two matrices... 

In regression, the equation A B.C is an approximation for example, A may represent a series of spectra diat are approximately equal to the product of two matrices such as scores and loadings matrices, hence this approach is important to obtain the best fit model for C knowing A and B or for B knowing A and C. 

The direct product of two matrices Q = P X R is a matrix whose dimensionality is the product of the dimensionalities of the two matrices. The components of Q consist of all products of the components of the separate matrices, P >Rm , with a convention as to ordering of the resultant components in Q. A specific example is given in Appendix D. 

The diagonal eigenvalue matrix L can be written as the product of two matrices T T in which the columns t( are mutually orthogonal and have their scalar product tjtj = X.j. If we let T be a (n x k) matrix which obeys these criteria, then T T = L. We can therefore write... 

Transpose, adjoint, and inverse of a matrix. The inverse has been defined in (4) above and the transpose and adjoint are defined in Appendix A.4-1 and Table 4-1.1. The reader is left to prove (see problem 4.1) that the transpose, adjoint, and inverse of the product of two matrices are given by ... 

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