Orthogonal Projector

(1)

Orthogonal Projector

Wanho Choi

(wanochoi.com)

(2)

Orthogonal Projector

x_a a = xa

x_a x

(3)

Orthogonal Projector

x_a a = xa

x

(4)

Orthogonal Projector

= a(aT_{x) ( ∵ a}T_{x : scalar)} x_a a = xa x_a x _x_a _{= (a ⋅ x)a = (a}T_x)a

(5)

Orthogonal Projector

= a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x x_a a = xa x x _x_a _{= (a ⋅ x)a = (a}T_x)a

(6)

Orthogonal Projector

x_a = (a ⋅ x)a = (aT_x)a = a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x = P_ax x_a a = xa x_a x

(7)

Orthogonal Projector

= a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x = P_ax ∴ P_a = aaT x_a a = xa x x _x_a _{= (a ⋅ x)a = (a}T_x)a

(8)

Orthogonal Projector

= a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x = P_ax ∴ P_a = aaT _{: projector} _{(rank = 1)} x_a a = xa x_a x _x_a _{= (a ⋅ x)a = (a}T_x)a

(9)

Orthogonal Projector

x_a = a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x = P_ax ∴ P_a = aaT _{(rank = 1)} b : projector x_b a = xa x x _x_a _{= (a ⋅ x)a = (a}T_x)a

(10)

Orthogonal Projector

x_a = a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x = P_ax ∴ P_a = aaT _{(rank = 1)} b x = x_a + x_b : projector x_b a = xa x_a x _x_a _{= (a ⋅ x)a = (a}T_x)a

(11)

Orthogonal Projector

x_a = a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x = P_ax ∴ P_a = aaT _{(rank = 1)} b x = x_a + x_b = (aaT_{)x + x} : projector x_b a = xa x_a x _x_a _{= (a ⋅ x)a = (a}T_x)a

(12)

Orthogonal Projector

x_a = a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x = P_ax ∴ P_a = aaT _{(rank = 1)} b x = x_a + x_b = (aaT_{)x + x} b ∴ x_b = x − (aaT_)x : projector x_b a = xa x_a x _x_a _{= (a ⋅ x)a = (a}T_x)a

(13)

Orthogonal Projector

x_a = a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x = P_ax ∴ P_a = aaT _{(rank = 1)} b x = x_a + x_b = (aaT_{)x + x} b ∴ x_b = x − (aaT_)x = (I − aaT_)x : projector x_b a = xa x_a x _x_a _{= (a ⋅ x)a = (a}T_x)a

(14)

Orthogonal Projector

x_a = a(aT_{x) ( ∵ a}T_{x : scalar)} = (aaT_)x = P_ax ∴ P_a = aaT _{(rank = 1)} b x = x_a + x_b = (aaT_{)x + x} b ∴ x_b = x − (aaT_)x = (I − aaT_{)x : complementary projector} : projector (rank = 2) x_b a = xa x_a x _x_a _{= (a ⋅ x)a = (a}T_x)a

(15)

Reflector

x x_a x_b b a x

(16)

Reflector

x_r = x − 2x_b x x_a x_b b a x_r

(17)

Reflector

x_r = x − 2x_b x x_a x_b b a x = x − 2(I − aaT_)x

(18)

Reflector

x_r = x − 2x_b x x_a x_b b a x_r = x − 2(I − aaT_)x = (I − 2(I − aaT_))x

(19)

Reflector

x_r = x − 2x_b x x_a x_b b a x = x − 2(I − aaT_)x = (I − 2(I − aaT_))x = (2aaT _{− I)x}

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Reflector

x_r = x − 2x_b x x_a x_b b a x_r = x − 2(I − aaT_)x = (I − 2(I − aaT_))x = (2aaT _{− I)x} ∴ P_r = 2aaT _{− I}

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