C++ Neural Networks and Fuzzy Logic: Preface

C++ Neural Networks and Fuzzy Logic

by Valluru B. Rao

MTBooks, IDG Books Worldwide, Inc.

ISBN: 1558515526   Pub Date: 06/01/95

Table of Contents

Appendix B

Mathematical Background

Dot Product or Scalar Product of Two Vectors

Given vectors U and V, where U = (u

1

, …, u

n

) and V = (v

1

, …, v

n

), their dot product or scalar product is U " V

= u

1

v

1

 +… + u

n

v

n

 = £ u

i

 v

i

.

Matrices and Some Arithmetic Operations on Matrices

A real matrix is a rectangular array of real numbers. A matrix with m rows and n columns is referred to as an

mxn matrix. The element in the ith row and jth column of the matrix is referred to as the ij element of the

matrix and is denoted by a

ij

.

The transpose of a matrix M is denoted by M

T

. The element in the ith row and jth column of M

T

 is the same as

the element of M in its jth row and ith column. M

T

 is obtained from M by interchanging the rows and columns

of M. For example, if

            2  7 −3              2  4

M =          , then M

T

 =  7  0

            4  0  9             −3  9

If X is a vector with m components, x

1

, …, x

m

, then it can be written as a column vector with components

listed one below another. It can be written as a row vector, X = (x

1

, …, x

m

). The transpose of a row vector is

the column vector with the same components, and the transpose of a column vector is the corresponding row

vector.

The addition of matrices is possible if they have the same size, that is, the same number of rows and same

number of columns. Then you just add the ij elements of the two matrices to get the ij elements of the sum

matrix. For example,

       3 −4 5    5 2 −3      8 −2  2

               +         =

       2  3 7    6 0  4      8  3 11

Multiplication is defined for a given pair of matrices, only if a condition on their respective sizes is satisfied.

Then too, it is not a commutative operation. This means that if you exchange the matrix on the left with the

matrix on the right, the multiplication operation may not be defined, and even if it is, the result may not be the

same as before such an exchange.

The condition to be satisfied for multiplying the matrices A, B as AB is, that the number of columns in A is

equal to the number of rows in B. Then to get the ij element of the product matrix AB, you take the ith row of

C++ Neural Networks and Fuzzy Logic:Preface

Appendix B Mathematical Background

416

---

A as one vector and the jth column of B as a second vector and do a dot product of the two. For example, the

two matrices given previously to illustrate the addition of two matrices are not compatible for multiplication

in whichever order you take them. It is because there are three columns in each, which is different from the

number of rows, which is 2 in each. Another example is given as follows.

                 3 −4 5              5  6

       Let A =            and   B =  2  0

                 2  3 7             −3  4

Then AB and BA are both defined, AB is a 2x2 matrix, whereas BA is 3x3.

                 −8  38              27 −2 67

       Also AB =         and BA  =    6 −8 10

                 −5  40              −1 24 13