Aside: projections onto vectors

1
Aside projections onto vectors

• If X? is an n?p data matrix and a a p?1 vector
  then Y? X?a is the projection of X? onto a
• Values of Y? give the coordinates of each
  observation along the vector a
• Example x1(1, 2)?, x2(2, 1)?, x3(?1,1)? a(3,
  ? 4)/5,
• So

2
Aside projections onto vectors
x1
2
?
x3
x2
?
?
(0,0)
?
1
3
Aside projections onto vectors
x1
2
?
x3
x2
?
?
(0,0)
?
1
a(3, ? 4)/5?
4
Aside projections onto vectors
x1
2
?
x3
x2
?
?
y1
(0,0)
?
1
a(3, ? 4)/5?
5
Aside projections onto vectors
x1(1, 2)?
2
?
x3
x2
?
?
y1 (3?1?4?2)/5 ?1
(0,0)
?
1
a(3, ? 4)/5?
6
Aside projections onto vectors
x1
2
?
x3
x2
?
?
y1 ?1
(0,0)
?
1
y2 2/5
a(3, ? 4)/5?
7
Aside projections onto vectors
x1
2
?
y3 ?7/5
x3
x2
?
?
y1 ?1
(0,0)
?
1
y2 2/5
a(3, ? 4)/5?
8
Aside projections onto vectors
2
y3 ?7/5
?
?
y1 ?1
(0,0)
?
1
?
y2 2/5
a(3, ? 4)/5?
9
Aside projections onto vectors
2
?
?
?
?
?
(0,0)
?
1
?
a(3, ? 4)/5?