43
where dY and dX are column vectors.
When X andY have the same order, the matrix
is square
and is called the Jacobian matrix of the transformation of X into Y,
and its determinant is called the Jacobian of the transformation.
d)
Derivative of the quadra

37
For example the quadratic polynomial equation
2
5x 2 + 6x 1 x2 + Sx 2 8
1
can be written as
(Note that A could be written
but that the cross product
coefficient Is split In two halves to make A symmetric).
An Important property of quadratic forms Is th

49
(cont 'd)
2)
h)
g)
0
0
0
k
0
0
[:
k
0
k
diagonal
b' c, e, f' g' h
d' f
scalar
e, f' g' h
co 1umn
b' h
identity
f' 9
none
I I)
1)
a' 9
none
a, d
square
2)
c, e
row
Answers:
ADDITION, MULTIPLICATION AND TRANSPOSITION (section 3 in notes)
From the followi

TEST RESULTS
The program was tested by computing the product matrix for the following
sequence of rotations and reflections:
o<
for the case
=
f3
=
f
For this example we have the following
input to ROTREF:
NUM = 10
NAXIS
= (3,
2, -1, 3, 2, -2, 3, 1, 2, 3

55
18)
forms.
Write the matrix of each of the following quadratic
What value class does each quadratic form belong to?
a)
5x 2 + 6xy + 5/
b)
5x
c)
7x
d)
5x
Answers:
2+
2
2
16)
4xy +
+ 8xy +
e)
2/
y
+ 2xy + 3y
f)
2
-5x
9l
g)
2
-2x
h)
Projective
a 11 but f.

62
SUMMARY OF REFLECTION AND ROTATION
1
Orthogonal Transformations
The matrix equation
Y
=A X
where A is a matrix and X and Y are column vectors, can be regarded as a
linear transformation, in which case the matrix A is CA.lled the
transformation matrix.

76
Heferences:
Carnahan, Luther and Wilkes (1969). "Applied Numerical Methods 11
Wiley. (page 334)
Fad.deev and Faddeeva (1963). "Computational Methods of Linear Algebra 11
Freeman. (page 144)
Thompson ( 1969). 11 Introduction to the Algebra of Matrices w

31
(213)
In the first case, the affine transformation can be reproduced
=
by a rotation (e
tan- 1 (t) and a stretching (k
transformation matrices
Rl =
sl
=
sl Rl
=
so that
Gos8 -sinJ
sine
case
=
=
'li3), having
[2/ffi m]
-3/
311T3
2/113
[: :] [7
[: -:1
an

MATRICES
I.
Introduction
Matrix notation is a powerful mathematical shorthand.
Concepts and relationships which often otherwise become buried
under a mass of symbols and equations, can in matrix notation be
expressed with brevity and clarity, leading to g

19
but the third equation is
2x 1 - sx 2 =
0
which are obviously inconsistent.
d)
Systems having a unique solution
If the rank of the coefficient matrix is equal to the
number of unknowns (the number of rows in the unknown vector X),
then there is one uni

25
- sine
R = [case
sine
J
(51 )
case
Illustrating the first interpretation (the vector is transformed) in
two dimensions:
From the diagram
xl
= r cos
x2
r sin
r cos case - r sin sine
yl = r cos (+8)
Y2
= r sin (+e)
=
r cos sine + r sin case
or
-s i neJ

7
c :]
A
5
and
3
B
=
5
4
then B = AT and A
9
Expressed in terms of their elements,
lbij=ajd
for
( 11 )
= 1 , 2, 3 and j = 1 , 2.
The transpose has the following properties
(AT)T = A
( 12)
(A + B) T = AT+ BT
( 13)
(nA) T = nAT
( 14)
(A B) T = BT AT (note r