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# Orthogonal Projector

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(1)

## Orthogonal Projector

(2)

xa a = xa

xa x

(3)

xa a = xa

x

(4)

### Orthogonal Projector

= a(aTx) ( ∵ aTx : scalar) xa a = xa xa x xa = (a ⋅ x)a = (aTx)a

(5)

### Orthogonal Projector

= a(aTx) ( ∵ aTx : scalar) = (aaT)x xa a = xa x x xa = (a ⋅ x)a = (aTx)a

(6)

### Orthogonal Projector

xa = (a ⋅ x)a = (aTx)a = a(aTx) ( ∵ aTx : scalar) = (aaT)x = Pax xa a = xa xa x

(7)

### Orthogonal Projector

= a(aTx) ( ∵ aTx : scalar) = (aaT)x = Pax ∴ Pa = aaT xa a = xa x x xa = (a ⋅ x)a = (aTx)a

(8)

### Orthogonal Projector

= a(aTx) ( ∵ aTx : scalar) = (aaT)x = Pax ∴ Pa = aaT : projector (rank = 1) xa a = xa xa x xa = (a ⋅ x)a = (aTx)a

(9)

### Orthogonal Projector

xa = a(aTx) ( ∵ aTx : scalar) = (aaT)x = Pax ∴ Pa = aaT (rank = 1) b : projector xb a = xa x x xa = (a ⋅ x)a = (aTx)a

(10)

### Orthogonal Projector

xa = a(aTx) ( ∵ aTx : scalar) = (aaT)x = Pax ∴ Pa = aaT (rank = 1) b x = xa + xb : projector xb a = xa xa x xa = (a ⋅ x)a = (aTx)a

(11)

### Orthogonal Projector

xa = a(aTx) ( ∵ aTx : scalar) = (aaT)x = Pax ∴ Pa = aaT (rank = 1) b x = xa + xb = (aaT)x + x : projector xb a = xa xa x xa = (a ⋅ x)a = (aTx)a

(12)

### Orthogonal Projector

xa = a(aTx) ( ∵ aTx : scalar) = (aaT)x = Pax ∴ Pa = aaT (rank = 1) b x = xa + xb = (aaT)x + x b ∴ xb = x − (aaT)x : projector xb a = xa xa x xa = (a ⋅ x)a = (aTx)a

(13)

### Orthogonal Projector

xa = a(aTx) ( ∵ aTx : scalar) = (aaT)x = Pax ∴ Pa = aaT (rank = 1) b x = xa + xb = (aaT)x + x b ∴ xb = x − (aaT)x = (I − aaT)x : projector xb a = xa xa x xa = (a ⋅ x)a = (aTx)a

(14)

### Orthogonal Projector

xa = a(aTx) ( ∵ aTx : scalar) = (aaT)x = Pax ∴ Pa = aaT (rank = 1) b x = xa + xb = (aaT)x + x b ∴ xb = x − (aaT)x = (I − aaT)x : complementary projector : projector (rank = 2) xb a = xa xa x xa = (a ⋅ x)a = (aTx)a

(15)

x xa xb b a x

(16)

### Reflector

xr = x − 2xb x xa xb b a xr

(17)

### Reflector

xr = x − 2xb x xa xb b a x = x − 2(I − aaT)x

(18)

### Reflector

xr = x − 2xb x xa xb b a xr = x − 2(I − aaT)x = (I − 2(I − aaT))x

(19)

### Reflector

xr = x − 2xb x xa xb b a x = x − 2(I − aaT)x = (I − 2(I − aaT))x = (2aaT − I)x

(20)

### Reflector

xr = x − 2xb x xa xb b a xr = x − 2(I − aaT)x = (I − 2(I − aaT))x = (2aaT − I)x ∴ Pr = 2aaT − I

(21)

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