Here we look at how to create a projection Matrix for orthogonal projection or transformation and projection onto a Line under Numerical Linear Algebra.

1. Projector And Projection Onto A Line
Numerical Linear Algebra
Isaac Amornortey Yowetu
NIMS-GHANA
July 10, 2020

2. Projection onto a Line Projection Matrix
Outline
1 Projection onto a Line
2 Projection Matrix

3. Projection onto a Line Projection Matrix
Graphical Example
Figure: A Graphical Example of a projection onto a line

4. Projection onto a Line Projection Matrix
Orthogonal Projection onto a line
Suppose we have L = span{u} is a line and v is projected onto
the line u, then ∃c ∈ R s.t:
ProjL(v) = c · u
v − ProjL(v) = v − c · u
We can obtain our orthogonal projection to be
(v − ProjL(v)) · (ProjL(v)) = (v − c · u) · (c · u) = 0 (1)
v(c · u) − (c · u)(c · u) = 0 (2)
(c · u)v = (c · u)(c · u) (3)
u · v = c(u · u) (4)

5. Projection onto a Line Projection Matrix
c =
u · v
u · u
Therefore
projL(v) =
u · v
u · u
u

6. Projection onto a Line Projection Matrix
Example
Let
v =


−2
3
−1

 and u =


−1
1
1


and let L be the line spanned by u. Compute projL(v) and
proj⊥
L (v)

7. Projection onto a Line Projection Matrix
Solution
projL(v) = (2)+(3)+(−1)
(1)+(1)+(1)


−1
1
1

 = 4
3


−1
1
1


proj⊥
L (v) = v − projL(v) =


−2
3
−1

 − 4
3


−1
1
1

 = 1
3


−2
5
−7



8. Projection onto a Line Projection Matrix
Projection Matrix
Here we would like to find a projection matrix P:
P = P2
and PT
= P
P =
uuT
uT u
projL(v) = Pv =
uuT
uT u
v
proj⊥
L (v) = v − Pv = v −
uuT
uT u
v

9. Projection onto a Line Projection Matrix
Example
Let
v =


−2
3
−1

 and u =


−1
1
1


and let L be the line spanned by u. Compute projL(v) and
proj⊥
L (v)

10. Projection onto a Line Projection Matrix
Solution
P =
uuT
uT u
u · uT
=


−1
1
1

 (−1 1 1) =


1 −1 −1
−1 1 1
−1 1 1


uT
· u = (−1 1 1)


−1
1
1

 = 3
∴ P =
1
3


1 −1 −1
−1 1 1
−1 1 1

 =


0.333 −0.333 −0.333
−0.333 0.333 0.333
−0.333 0.333 0.333



11. Projection onto a Line Projection Matrix
projl (v) = Pv =


0.333 −0.333 −0.333
−0.333 0.333 0.333
−0.333 0.333 0.333




−2
3
−1


=
4
3


−1
1
1


proj⊥
L (v) = v − projL(v) =


−2
3
−1

 − 4
3


−1
1
1

 = 1
3


−2
5
−7

