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
6
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
7
.
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