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Projections

Module by: Mark A. Davenport. E-mail the author

Definition 1

A linear transformation P:X→XP:X→X is called a projection if P(x)=xP(x)=x∀x∈R(P)∀x∈R(P), i.e, P(P(x))=P(x)P(P(x))=P(x)∀x∈X.∀x∈X.

Example 1

P:R3→R3P:R3→R3, P(x1,x2,x3)=(x1,x2,0)P(x1,x2,x3)=(x1,x2,0)

Figure 1
An illustration showing the projection operator that maps x to the x_1 x_2 - plane.

Definition 2

If PP is a projection operator on an inner product space VV, we say that PP is an orthogonal projection if R(P)⊥N(P)R(P)⊥N(P) , i.e., 〈x,y〉=0〈x,y〉=0∀x∈R(P),y∈N(P).∀x∈R(P),y∈N(P).

If PP is an orthogonal projection, then for any x∈Vx∈V we can write:

x = P x + ( I - P ) x x = P x + ( I - P ) x
(1)

where Px∈R(P)Px∈R(P) and (I-P)x∈N(P)(I-P)x∈N(P) (since P(I-P)x=Px-P(Px)=Px-Px=0P(I-P)x=Px-P(Px)=Px-Px=0.)

Now we see that the solution to our “best approximation in a linear subspace” problem is an orthogonal projection: we wish to find a PP such that R(P)=AR(P)=A.

Figure 2
(a) (b)
Illustrations of the projection operator for a 1-D subspace of R2 and a 2-D subspace of R3. The best approximation to a point from the subspace is given by a projection of the point into that subspace.Figure 2(b) (06_3.png)

The question is now, how can we design such a projection operator?

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