**Orthogonal Projector**

**Wanho Choi**

**(wanochoi.com)**

**Orthogonal Projector**

**x _{a}**

**a =**

**xa**

**x _{a}**

**x**

**Orthogonal Projector**

**x _{a}**

**a =**

**xa**

**x**

**Orthogonal Projector**

**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**x**

_{a}**a =**

**xa**

**x**

_{a}**x**

_{x}_{a}

_{= (a ⋅ x)a = (a}*T*

_{x)a}**Orthogonal Projector**

**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**x**

_{a}**a =**

**xa**

**x**

**x**

_{x}_{a}

_{= (a ⋅ x)a = (a}*T*

_{x)a}**Orthogonal Projector**

**x**

_{a}**= (a ⋅ x)a = (a**

*T*

_{x)a}**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**= P**

_{a}x**x**

_{a}**a =**

**xa**

**x**

_{a}**x**

**Orthogonal Projector**

**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**= P**

_{a}x**∴ P**

_{a}**= aa**

*T*

**x**

_{a}**a =**

**xa**

**x**

**x**

_{x}_{a}

_{= (a ⋅ x)a = (a}*T*

_{x)a}**Orthogonal Projector**

**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**= P**

_{a}x**∴ P**

_{a}**= aa**

*T*

_{: projector}

_{(rank = 1)}**x**

_{a}**a =**

**xa**

**x**

_{a}**x**

_{x}_{a}

_{= (a ⋅ x)a = (a}*T*

_{x)a}**Orthogonal Projector**

**x**

_{a}**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**= P**

_{a}x**∴ P**

_{a}**= aa**

*T*

_{(rank = 1)}**b**

*: projector*

**x**

_{b}**a =**

**xa**

**x**

**x**

_{x}_{a}

_{= (a ⋅ x)a = (a}*T*

_{x)a}**Orthogonal Projector**

**x**

_{a}**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**= P**

_{a}x**∴ P**

_{a}**= aa**

*T*

_{(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}**Orthogonal Projector**

**x**

_{a}**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**= P**

_{a}x**∴ P**

_{a}**= aa**

*T*

_{(rank = 1)}**b**

**x = x**

_{a}**+ x**

_{b}**= (aa**

*T*

_{)x + x}*: projector*

**x**

_{b}**a =**

**xa**

**x**

_{a}**x**

_{x}_{a}

_{= (a ⋅ x)a = (a}*T*

_{x)a}**Orthogonal Projector**

**x**

_{a}**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**= P**

_{a}x**∴ P**

_{a}**= aa**

*T*

_{(rank = 1)}**b**

**x = x**

_{a}**+ x**

_{b}**= (aa**

*T*

_{)x + x}**b**

**∴ x**

_{b}**= x − (aa**

*T*

_{)x}*: projector*

**x**

_{b}**a =**

**xa**

**x**

_{a}**x**

_{x}_{a}

_{= (a ⋅ x)a = (a}*T*

_{x)a}**Orthogonal Projector**

**x**

_{a}**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**= P**

_{a}x**∴ P**

_{a}**= aa**

*T*

_{(rank = 1)}**b**

**x = x**

_{a}**+ x**

_{b}**= (aa**

*T*

_{)x + x}**b**

**∴ x**

_{b}**= x − (aa**

*T*

_{)x}**= (I − aa**

*T*

_{)x}*: projector*

**x**

_{b}**a =**

**xa**

**x**

_{a}**x**

_{x}_{a}

_{= (a ⋅ x)a = (a}*T*

_{x)a}**Orthogonal Projector**

**x**

_{a}**= a(a**

*T*

_{x) ( ∵ a}*T*

_{x : scalar)}**= (aa**

*T*

_{)x}**= P**

_{a}x**∴ P**

_{a}**= aa**

*T*

_{(rank = 1)}**b**

**x = x**

_{a}**+ x**

_{b}**= (aa**

*T*

_{)x + x}**b**

**∴ x**

_{b}**= x − (aa**

*T*

_{)x}**= (I − aa**

*T*

_{)x : complementary projector}*: projector*

*(rank = 2)*

**x**

_{b}**a =**

**xa**

**x**

_{a}**x**

_{x}_{a}

_{= (a ⋅ x)a = (a}*T*

_{x)a}**Reflector**

**x**

**x**

_{a}**x**

_{b}**b**

**a**

**x**

**Reflector**

**x**

_{r}**= x − 2x**

_{b}**x**

**x**

_{a}**x**

_{b}**b**

**a**

**x**

_{r}**Reflector**

**x**

_{r}**= x − 2x**

_{b}**x**

**x**

_{a}**x**

_{b}**b**

**a**

**x**

**= x − 2(I − aa**

*T*

_{)x}**Reflector**

**x**

_{r}**= x − 2x**

_{b}**x**

**x**

_{a}**x**

_{b}**b**

**a**

**x**

_{r}**= x − 2(I − aa**

*T*

_{)x}**= (I − 2(I − aa**

*T*

_{))x}**Reflector**

**x**

_{r}**= x − 2x**

_{b}**x**

**x**

_{a}**x**

_{b}**b**

**a**

**x**

**= x − 2(I − aa**

*T*

_{)x}**= (I − 2(I − aa**

*T*

_{))x}**= (2aa**

*T*

_{− I)x}**Reflector**

**x**

_{r}**= x − 2x**

_{b}**x**

**x**

_{a}**x**

_{b}**b**

**a**

**x**

_{r}**= x − 2(I − aa**

*T*

_{)x}**= (I − 2(I − aa**

*T*

_{))x}**= (2aa**

*T*

_{− I)x}**∴ P**

_{r}**= 2aa**

*T*

_{− I}