Page 308 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Since
A ( f , )
g ( ) = jk k n
j n
k
A jk (M *) kn
=
k
=
jn
it follows that E( ) = for 1 n r. This implies E = for every in W. Therefore, E maps V
n n
onto W and E = E. If is an arbitrary vector in V, then
2
f( , E ) = f n g j ( ) j
n
j
g ( ) ( , a )
f
= j n j
j
= A jk ( f k , ) f ( n , j )
j k
–1
Since A* = M , it follows that

f( , E ) = (M 1 ) M jn ( f k , )
kj
n
k j
( f , )
= kn k
k
= f( , ).
n
This implies f( , E ) = f( , ) for every in W. Hence
f( , – E ) = 0
for all in W and in V. Thus 1 – E maps V into W’. The equation
= E + (1 – E)

shows that V = W + W’. One final point should be mentioned. Since W W’ = {0}, every vector
in V is uniquely the sum of a vector in W and a vector in W’. If is in W’, it follows that E = 0.
Hence I – E maps V onto W’.
The projection E constructed in the proof may be characterized as follows: E = if and only if
is in W and — belongs to W’. Thus E is independent of the basis of W that was used in its
construction. Hence we may refer to E as the projection of V on W that is determined by the
direct sum decomposition

V = W W’.
Note that E is an orthogonal projection if and only if W’ = W .
Theorem 4: Let f be a form on a real or complex vector space V and A the matrix of f in the ordered
basis { , ..., } of V. Suppose the principal minors of A are all different from 0. Then there is a
1 n
unique upper triangular matrix P with P = 1 (1 k n ) such that
kk
P*AP
is upper-triangular.

302 LOVELY PROFESSIONAL UNIVERSITY