Mathematical Fundamentals

123

unit diagonal matrix

inverse matrix

power

There is a concise way to assign to a variable all the values regularly spaced

(with step inc) between a min and a max:

x = [min, inc, max];

This kind of instruction has been used to plot the function of fig. 2. After having

defined the domain as the vector of points

l=[0.5: 0.1: 4.0];

the vector representing the codomain has been computed by application of the

function to the vector l:

f=1./(2*l)*sqrt(t/r);
A.4.1

Square Matrices

The n-th order square matrices defined over a field F are a set F

n×n

which

is very important for its affinity with the classes of numbers. In fact, for these

matrices the sum and product are always defined and it is easy to verify that

the properties S14, P1, and D12 of appendix A.1 do hold. The property P3

is also verified and the neutral element for the product is found in the unit

diagonal matrix, which is a matrix that has ones in the main diagonal
and zeros

elsewhere. In general, the commutativity is not ensured for the product, and a

matrix might not admit an inverse matrix, i.e., an inverse obeying to property

P4. In the terminology introduced in appendix A.1, the square matrices F
n×n

form a ring with a unity. This observation allows us to treat the square matrices

with compact notation, as a class of numbers which is not much different from

that of integers
A.5

Exponentials and Logarithms

Given a number a R

+

, it is clear what is its natural m-th power, that is

the number obtained multiplying a by itself m times. The rational power a

1/m

,

with m a natural number, is defined as the number whose m-th power gives

a. If we extend the power operator to negative exponents by reciprocation of

the positive power, we give meaning to all powers a

r

, with r being any rational

number. The extension to any real exponent is obtained by imposing continuity

to the power function. Intuitively, the function f (x) = a

x

describes a continuous

curve that "interpolates" the values taken at the points where x is rational. The

power operator has the following fundamental properties:

E1 : a

x

a

y

= a

x+y

E2 :

a

x

a

y

= a

x-y

E3 : (a

x

)

y

= a

xy

E4 : (ab)

x

= a

x

b

x

.

6

The main diagonal goes from the top leftmost corner to the bottom rightmost corner.

7

Two important differences with the ring of integers is the non commutativity and the

possibility that two non-zero matrices multiplied together give the zero matrix (the zero

matrix admits non-zero divisors).