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3. Numerical Linear Algebra 95
4. PROJECTION MATRIX
4.1 Projection of the Vector ‘a’ on the Vector ‘b’
Figure 3-4. Projection Illustration
Consider two vectors a and b. The projection of the vector ‘a’ on ‘b’ is the
scaled version of the vector ‘b’ (i.e) Pb. The vector e is orthogonal to the
vector b as shown in the figure 3-3.
e=a-Pb
T
⇒ b (a-Pb) =0
T
T
⇒ b a =Pb b
T
T
⇒P = (b a) / (b b)
T
bP = [b( b a)
Therefore the projection of the vector ‘a’ on ‘b’ is given as ] /
T
[(b b)].
The Projection matrix (In this case, ,it is the single element matrix) is
T
T
given as (bb )/( b b) = [P M]
Thus the projection of the vector ‘a’ on ‘b’ is given as [P M] a
