Page 348 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes preserve f, the product ST also preserves f; for f(ST , ST ) = f(T , T ) = f( , ). In other words the
collection of linear operators which preserve f, is closed under the formation of operator products.
Consider a bilinear form given by

n
= a x y
ij i j
, i j 1
If we introduce

x 1 y 1
x 2 y 2
X = , Y
 
x y
n n
then

t
B = X AY
where n rowed square matrix A is
A = [a ]
ij
In case Y = X then we have a quadratic form

n n
Q = a x x
ij i j
i 1 j 1
In matrix form

Q = X AX
T
We now consider certain transformation operator P such that
X = PX’
where P is non-singular (or invertible), then

t
t
t
X = (PX’) = X’ P t
So
t
t
Q = X’ P APX’
Defining
t
A’ = P AP
We have
t
Q = X’ A’X’
If A is symmetric then
t
t
t
t
t
A ’ = (P AP) = P A P = P AP = A’
t
Thus symmetry of the matrix is maintained. Now if Q represents the length of the vector (x , x ,
1 2
... x ) then preservation of length means;
n
t
t
X X = X ’P PX’ = X ’X’, if
t
t
t
PP = I
which means that P is an orthogonal matrix.
One of the examples of the orthogonal transformation the rotation of co-ordinate system.
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