Addition and multiplication of matrices

If a matrix A and a number c are given, the multiplication of the matrix can be defined as 

Process engineering and design using Visual Basic 

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First we present the rules for equality, addition, and multiplication of matrices. Equality... 

Given a representation JP of , associating s with D(s), we can form a representation of the algebra by associating to each element a the matrix D(a) — 2 a(s) D(s). It is evident that addition and multiplication of the matrices so defined will reproduce the rules (7) and (8). 

Addition and subtraction of matrices are performed element-wise the matrices must have the same size (see Figure A.2.2). Multiplication of a vector or a matrix with a scalar is also performed element-wise in the case of a vector the resulting vector has the same direction but a different length. Vectors a and b = —a have the same length but reverse direction. 

The algebra of multi-way arrays is described in a field of mathematics called tensor analysis, which is an extension and generalization of matrix algebra. A zero-order tensor is a scalar a first-order tensor is a vector a second-order tensor is a matrix a third-order tensor is a three-way array a fourth-order tensor is a four-way array and so on. The notions of addition, subtraction and multiplication of matrices can be generalized to multi-way arrays. This is shown in the following sections [Borisenko Tarapov 1968, Budiansky 1974],... 

Problem 61-1. Show that the laws of ordinary algebra hold for the addition and subtraction of matrices and their multiplication by scalare for example,... 

FIGURE A.2.2 Addition of matrices and multiplication of a matrix with a scalar. 

In this section we explore the matrix analogues of addition, subtraction and multiplication of numbers. The analogue for division (the inverse operation of multiplication) has no direct counterpart for matrices. 

Gabriel KR, Least squares approximation of matrices by additive and multiplicative models, Journal of the Royal Statistical Society Series B Statistical Methodology, 1978, 40, 186-196. 

Matrices in the form, combined by matrix addition and multiplication, are isomorphic with the field of complex numbers (a + This way. 

Third, all manipulations required for the density-hased optimization of the Hartree-Fock energy may be carried out as matrix additions and multiplications. Owing to the sparsity of the matrices, the cost of these manipulations scales linearly with the size of the system. 

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... 

Since it is necessary to represent the various quantities by vectors and matrices, the operations for the MND that correspond to operations using the univariate (simple) Normal distribution must be matrix operations. Discussion of matrix operations is beyond the scope of this column, but for now it suffices to note that the simple arithmetic operations of addition, subtraction, multiplication, and division all have their matrix counterparts. In addition, certain matrix operations exist which do not have counterparts in simple arithmetic. The beauty of the scheme is that many manipulations of data using matrix operations can be done using the same formalism as for simple arithmetic, since when they are expressed in matrix notation, they follow corresponding rules. However, there is one major exception to this the commutative rule, whereby for simple arithmetic ... 

The algebra of matrices gives rules for (1) equality, (2) addition and subtraction, (3) multiplication, and (4) division as well as (5) an associative and a distributive law. It also includes definitions of (6) a transpose, adjoint and inverse of a matrix. 

So long as dimensional conformability is maintained, such super-matrices (matrices of matrices) obey additive, scalar-multiplicative, and matrix-multiplicative equations analogous to (9.8)—(9.11), such as... 

The basic matrix operations +, — and correspond to the normal matrix addition, subtraction and multiplication (using the dot product) for scalars these are also defined in the usual way. For the first two operations the two matrices should generally have the same dimensions, and for multiplication the number of columns of die first matrix should equal the number of rows of die second matrix. It is possible to place the results in a target or else simply display them on die screen as a default variable called ans. 

Addition of two square matrices P + R is performed by taking the sums of corresponding elements, Pmn + Rmn. Multiplication of two square matrices P and R to give a square matrix Q is defined by... 

The rules of addition and scalar multiplication obviously give, for any number of matrices of the same order mxn ... 

Next we briefly discuss the diagonal, 9-dependent, elements of the 3x3 A-matrix as presented in Figs. l(c,f) and the listed 3x3 D-matrix diagonal elements. Both, the A-and the D-matrices become important if and only if the a-values, as calculated for the two-state system, are not multiples of n (or zero). In the case where all relevant a-values of the system are multiples of n (or zero), the j-th A-matrix diagonal element is expected to be close either to cos(7jj+i(9)) or to cos(7j ij(9)), as the case may be, and therefore the diagonal D-matrix elements are expected to be equal to 1. As is noticed, the results due to both matrices confirm the two-state results (and do not yield any additional information). For instance the absolute value of all the D-matrix diagonal elements is, indeed, - 1 moreover, in the first case (Fig. 1(c)), the (1,1) and the (2,2) elements are equal to -1, which implies that we encounter a (1,2) ci, and in the second case, the (2,2) and the (3,3) elements are equal to -1, which implies that a (2,3) ci is encountered (for a detailed analysis on this subject see Ref. 22). 

Now, the multiplication of two square matrices of order m involves multiplication operations and — 1) additions. However, the multiplication of a square and a column... 

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