Matrices products

Chapter 9 dealt with the basic operations of addition of two matrices with the same dimensions, of scalar multiplication of a matrix with a constant, and of arithmetic multiplication element-by-element of two matrices with the same 

Note that the inner dimensions of X and Y must be equal. For this reason the operation is also called inner product, as the inner dimensions of the two terms vanish in the product. Any element of the product, say can also be thought of as 

Throughout the book, matrices are often subscripted with their corresponding dimensions in order to provide a check on the conformity of the inner dimensions of matrix products. For example, when a 4x3 matrix X is multiplied with a 3x2 matrix Y rows-by-columns, we obtain a 4x2 matrix Z  

In this theoretical chapter, however, we do not follow this convention of subscripted matrices for the sake of conciseness of notation. Instead, we will take care to indicate the dimensions of matrices in the accompanying text whenever this is appropriate. 

The operation of matrix multiplication can be shown to be associative, meaning that X(YZ) = (XY)Z. But, it is not commutative, as in general we will have that XY YX. Matrix multiplication is distributive with respect to matrix addition, which implies that (X + Y)Z = XZ + YZ. When this expression is read from right to left, the process is called factoring-out [4]. 

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It is helpful to remember that the element py is formed from the ith row of the first matrix and the jth column or the second matrix. The matrix product is not commutative. That is, AB BA in general. 

The r vectors are the principal axes of inertia determined by diagonalization of the matrix of inertia (eq. (12.14)). By forming the matrix product P FP, the translation and rotational directions are removed from the force constant matrix, and consequently the six (five) trivial vibrations become exactly zero (within the numerical accuracy of the machine). 

The proof takes different forms in different representations. Here we assume that quantum states are column vectors (or spinors ) iji, with n elements, and that the scalar product has the form ft ip. If ip were a Schrodinger function, J ftipdr would take the place of this matrix product, and in Dirac s theory of the electron, it would be replaced by J fttpdr, iji being a four-component spinor. But the work goes through as below with only formal changes. Use of the bra-ket notation (Chapter 8) would cover all these cases, but it obscures some of the detail we wish to exhibit here. 

Tr designating the trace of the matrix product that follows. This is seen on direct expansion. Also, if a is a statistical matrix belonging to a certain sharp value of a specified observable, the probability that a system with matrix p shall exhibit that value upon measurement is... 

Being a product of delta functions, it is an improper expression, and the trace is undefined. However, it is useful to consider the operator product P(X)P where R is any linear operator defined in configuration space. The typical element of the matrix product is... 

For this purpose, let us use invariance of the matrix product trace under cyclic permutation of factors and represent (7.11) as... 

Now the pairs of rotation matrix products in Eq. (A.7) can be replaced with Clebsch-Gordan series... 

Polymer Matrix- Substrate and Matrix- Products Compatibility... 

An important property of the matrix product is that the transpose of a product is equal to the product of the transposed terms in reverse order ... 

Matrix multiplication can be applied to vectors, if the latter are regarded as one-column matrices. This way, we can distinguish between four types of special matrix products, which are explained below and which are represented schematically in Fig. 29.6. 

The vector-by-matrix product involves an n vector which premultiplies an nxp matrix Y to yield a p vector z ... 

Fig. 29.6. Schematic illustration of four types of special matrix products the matrix-by-vector product, the vector-by-matrix product, the outer product and the scalar product between vectors, respectively from top to bottom.

The outer product of two vectors can be thought of as the matrix product between a single-column matrix with a single-row matrix ... 

From V and A one can reconstruct the original matrix A by working out the consecutive matrix products ... 

The matrix-to-vector product can be interpreted geometrically as a projection of a pattern of points upon an axis. As we have seen in Section 29.4 on matrix products, if X is an nxp matrix and if v is a p vector then the product of X with v produces the n vector s ... 

The least squares solution of MLR can be formally defined in terms of matrix products (Section 10.2) ... 

An important theorem of matrix algebra, called singular value decomposition (SVD), states that any nxp table X can be written as the matrix product of three terms U, A and V ... 

Algebraically, the reconstruction of the values of X has been defined by the matrix product of the scores S with the transpose of the loadings L (eq. (31.22)). Geometrically, one reconstructs the value Xy by perpendicular projection of the point represented by upon the axis represented by s, as shown in Fig. 31.3c ... 

In NIPALS one starts with an initial vector t with n arbitrarily chosen values (Fig. 31.12). In a first step, the matrix product of the transpose of the nxp table X with the n-vector t is formed, producing the p elements of vector w. Note that in the traditional NIPALS notation, w has a different meaning than that of a weighting vector which has been used in Section 31.3.6. In a second step, the elements of the p-vector w are normalized to unit sum of squares This prevents values from becoming too small or too large for the purpose of numerical computation. The... 

In the power algorithm one first computes the matrix product of Cp with an initial vector of p random numbers v, yielding the vector w ... 

A contingency table X can be constructed by means of the matrix product of two indicator tables ... 

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