Addition and subtraction of matrices

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... 

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

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. 

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... 

Addition and Subtraction The operations of addition (+) and subtraction (-) of two or more matrices are possible if and only if they have the same number of rows and columns. Thus A B = (iay by) i.e., addition and subtraction are of corresponding elements. 

Addition or subtraction of matrices is done element-wise and thus is straightforward. Obviously, the dimensions of the matrices to be added have to match. 

Equality of Matrices Let A = (ay), B = (by). Two matrices A and B are equd (=) if and only if they are identical that is, they have the same number of rows and the same number of columns and equal corresponding elements ay = by for all i andj)-Addition and Subtraction The operations of addition (+) and subtraction (-) of two or more matrices are possible if and only if they have the same number of rows and columns. Thus A B = ay by) i.e., addition and subtraction are of corre onding elements. 

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,... 

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 ... 

Addition or subtraction of two matrices (both must contain the same number of rows and columns) ... 

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],... 

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. 

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. 

Matrix addition/subtraction Two matrices A and B are said to be compatible for addition/sub-traction if both the matrices have the same numbers of rows and columns. The addition/subtraction of two matrices is carried out elementwise and simply follows... 

Data types treated by MADCAP include real numbers (no distinction is made between floating-point and integer), complex numbers, character strings, matrices with real elements, sets of natural numbers, and arrays of arbitrary elements (77). Notation and facilities exist for character-string manipulation matrix operations of addition, subtraction, multiplication, inversion, and transposition set-theoretic operations of complementation, union, intersection, subtraction, and symmetric subtraction (exclusive or). When the set-theoretic operations were added to the language, set-theoretic comparators for use in conditional expressions and other language extensions to facilitate the programming of combinatorial calculations were incorporated, as well (78). 

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 ... 

An algebra typically involves the operations of adding, subtracting, multiplying, or dividing the objects it describes, whether matrices or simple numbers. For completeness, we now summarize some other aspects of matrix algebra, built on the fundamental definitions of addition/subtraction (9.8), scalar multiplication (9.9), and matrix multiplication (9.11). 

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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