The Product of Two Matrices

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 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... 

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... 

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. 

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 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 ... 

A basic condition for the product of two matrices is that the number of columns of the left matrix and rows of the right matrix are equal (m). 

Since the rank of Y can be estimated by factor-analytical technique with consideration of experimental noise, the number of unexpected interferents, M, can be obtained easily by subtracting N from the rank of Y. The information on the number of interferents is crucial in this situation, this makes the distinction between matrix calibration and vector calibration. Assuming the bilinear structure of the response, one can factor-decompose the overall background responses of M interferents into the product of two matrices... 

However, the largest eigenvalue is not itself a norm because the largest eigenvalue of the product of two matrices can be larger than the product of... 

We can reexpress this matrix as the product of two matrices. The first is a circulant matrix with every row29 and every column a copy of the mask divided by 2. The second is a diagonal matrix with the value 2 on the diagonal of slope -2. We call the latter the sampling matrix and the former the smoothing matrix. 

Determining the inverse of a matrix by hand is a fairly complicated matter. Fortunately, Excel has a built-in function, MINVERSE, that will perform the inversion. It also has a matrix multiplication function, MMULT, that will calculate the product of two matrices. In order to let the spreadsheet know that your instructions concern an entire block or array rather than an individual cell, these two functions require that you first highlight the entire block to which the instruction applies, and then enter the instruction while simultaneously depressing Ctrl, Shift, and Enter. 

Computes the product of two matrices Mout = Ml times M2 rl number of rows in Ml cl number of columns in Ml... 

Thus, the determinant of the product of two matrices is equal to the product of their determinants. 

An example of such a matrix was found in Example 2.3. All rotation matrice.s are orthogonal and have real determinants. Since the determinant of the product of two matrices is equal to the product of the two determinants, and because the detemtinant of a matrix and its transpose are equal, the determinant of a real orthogonal matrix must equal 1. For a unitary matrix, U,... 

All direct methods for solution of the system of linear equations (3) are based on the decomposition of matrix N into the product of two matrices. 

The statistical weight matrix for bond i, denoted U , is usually formulated as the product of two matrices. 

For this reason, it is necessary to use terminology which specifies the order of multiplication. For example, if the matrices A and B are n x n, then C = AB is read as B premultiplied by A whereas D = BA is read as B postmultiplied by A. Fiuther, the results AB and BA are not necessarily equal. This is a very different phenomenon from the usual symbolic algebra and may even result in the product of two matrices being the null matrix without either matrix being nuU. For instance. 

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