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 ... 

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. 

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

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. 

Given two equal-sized graph-theoretical matrices Mj and M2, difference matrices, denoted as aj-e obtained by subtracting the corresponding elements of matrices Mi and M2 ... 

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... 

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],... 

Real matrices should be used which are similar to the unknown sample. First, the level of the incurred (native) compound(s) must be accurately evaluated before spiking, and the percentage recovery is usually referred to as the sum of original content plus the spike or directly to the spike amount after subtraction of the original content. 

The advantage of matrix representation consists of the fact that all of the matrix operations can be applied to the encoded chemical structures. Simple subtraction of two BE matrices provides. 

If two rows or colunms are identical except for a multiplicative constant, the determinant is zero. This is easily seen, since one of these rows/columns can be made into a zero-vector by subtraction of the two, and expansion according to this zero row/column by eq. (16.18) will give zero. Such matrices may arise owing to linear dependencies of the rows or colums. 

Thus, addition (or subtraction) of two matrices is accomplished by adding (or subtracting) the corresponding elements of the two matrices. These operations may, of course, be extended to any number of matrices of the same order. In particular, observe that adding k identical matrices A (equivalent to scalar multiplication of A by k) results in a matrix in which each element is merely the corresponding element of A multiplied by the scalar k. Thus, when a matrix is to be multiplied by a scalar, the required operation is done by multiplying each element of the matrix by the scalar. For example, for A given in relation (A.2),... 

Application of matrices with the supported metals as catalysts enabled us to attain rather high content of carbon nanostructures (20-30% of the process produet). Content of nanotubes in pyrolytie deposition was determined by subtraction of the mass of amorphous carbon from the mass of resulting product (amorphous carbon plus nanotubes). According to electron microscopic data, in the methods of obtaining of carbon nanostructures with reduction nanoparticles of metals in atmosphere of hydrogen a lot of nanotubes were formed with external diameters in the range 35-52 nm for the cobalt catalyst... 

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. 

---