Matrices addition and subtraction

Matrix algebra also involves the additum and subtraction of matrices. The rules for this are as follows  

This is defined only for matrices of exactly the same shape (same number of rows and columns). For addition of two mxn matrices A and B, the sum C is given by 

If o is a scalar, then the operation gA involves multiplying every element of A by a. 

The sum of two matrices is obtained by adding the corresponding elements of the two matrices. For example, given 

Note that the resulting matrix has the same dimensions as the original matrices two 

The negative -A of a matrix A is simply the matrix whose elements are the negatives of the corresponding elements of A. 

The difference between two matrices is obtained by subtracting the corresponding elements of the second matrix from the elements of the first. For example, given the two matrices A and B on the previous page, their difference is 

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Use the evalm command to perform matrix operations. For example, matrix addition and subtraction can be done ... 

NOTE Because A. B is carried out on elemental level, it requires that both A and B have the same size (see Sec. 2.2). Matrix addition and subtraction are defined based on an element-by-element array operation. 

Addition and subtraction of matrices is canied out by adding or subtracting coiTcsponding elements. With matrices denoted by boldface capital letters and matrix elements by lower case letters, if... 

These are solved by matrix manipulations. Programs are in POLYMATH, CONSTANTINIDES AND CHAPRA CANALE. When the number of equations is not large, a manual procedure can be used to eliminate one variable at a time by reduction of the leading coefficients to unity and appropriate additions and subtractions of the equations. 

In Excel, mathematical operations of one or more cells can be dragged to other cells. Since a cell represents one element of an array or matrix, the effect will be an element-wise matrix calculation. Thus, addition and subtraction of matrices are straightforward. An example ... 

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. 

In addition techniques, the test substance concentration is determined from the difference in the ISE potentials obtained before and after a change in the sample solution concentration. The main advantage Ues in the fact that the whole measurement is carried out in the presence of the sample matrix, so that results with satisfactory accuracy and precision can be obtained even if a substantial portion of the test substance is complexed. Several addition techniques can be used, namely, single, double or multiple known addition methods, in which the sample concentration is increased by additions of a test substance standard solution single, double or multiple known subtraction methods, in which the sample concentration is decreased by additions of a standard solution of a substance that reacts stoichiometrically with the determinand and analyte addition and subtraction methods, in which the sample is added to a test substance solution or to a reagent solution. 

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. 

Addition and subtraction. Matrices may be added and subtracted only if they are of the same dimensions. Under these circumstances, the sum of A and B is given by the matrix C,... 

Addition and subtraction is the most straightforward operation. Each matrix (or vector) must have die same dimensions, and simply involves performing die operation element by element. Hence... 

These rules for adding and subtracting matrices give matrix addition the same properties as ordinary addition and subtraction. It is closed (among matrices of the same size), commutative, and associative. There is an additive identity (the matrix consisting entirely of zeros) and an additive inverse ... 

Fig. 5.3.13 Hadamard encoding and decoding for simultaneous four-slice imaging. The encoding is based on four experiments, A-D. In each experiment, all four slices are excited by a multi-frequency selective pulse. Its phase composition is determined by the rows of the Hadamard matrix H2. The image response is the sum of responses for each individual, frequency selective part of the pulse. Thus, addition and subtraction of the responses to the four experiments separates the information for each slice. This operation is equivalent to Hadamard transformation of the set of image responses. Adapted from [Miil21 with permission from Wiley-Liss. Inc., a division of John-Wiley Sons, Inc.

There is another notation that can be used to write the antisymmetrized wave function of Eq. (18.6-5). It is known as Slater determinant. Adetermhiaiitis a quantity derived from a square matrix by a certain set of multiplications, additions, and subtractions. There is a brief introduction to matrices and determinants in Appendix B. If the elements of the matrix are constants, the determinant is equal to a single constant. If the elements of the matrix are orbitals, the determinant of that matrix is a single function of all of the coordinates on which the orbitals depend. The wave function of Eq. (18.6-5) is equal to the determinant ... 

The transpose of a matrix M is calculated using the command M. The inverse of a matrix M is calculated as inv(M). In MATLAB, the multiphcation of matrices A and B is denoted by A B, while their addition and subtraction are denoted by A -h B and A - B, respectively. Commands for element-by-element multiphcation and division are also available. For more functions and help on any MATLAB operation, type help. 

Matrix C is also a (3 x 2) matrix. The commutative and associative laws for addition and subtraction apply. 

The following important rules of tensor analysis should be mentioned Addition and subtraction is only defined for tensors of the same rank and of the same transformation properties (co-/contravariance). For example, adding a matrix and a vector is not valid. Multiplication (also called tensor contraction) is only defined for pairs of indices, where one index is co- and the other is contravariant. As another example, odxi is a valid tensor contraction, but x x is not. 

The normal rules of association and commutation apply to addition and subhaction of matrices just as they apply to the algebra of numbers. The zero matrix has zero as all its elements hence addition to or subtraction from A leaves A unchanged... 

A vector can be thought of as a point in -dimensional space, although the graphical representation of such a point, when the dimension of the vector is greater than 3, is not feasible. The general rules for matrix addition, subtraction, and multiplication described in Section A.2 apply also to vectors. 

Let s look at matrix addition, subtraction, multiplication by scalars, and matrix multiplication (multiplication of a matrix by a matrix). 

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. 

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