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Chapter 2 Projection Matrices 2.1 Definition Definition 2.1 Let x ∈ E n = V ⊕W. Then x can be uniquely decomposed into x = x 1 +x 2 (where x 1 ∈ V and x 2 ∈ W). The transformation that maps x into x 1 is called the projection matrix (or simply projector) onto V along W and is denoted as φ. This is a linear transformation; that is, φ(a 1 y 1 +a 2 y 2 ) = a 1 φ(y 1 ) +a 2 φ(y 2 ) (2.1) for any y 1 , y 2 ∈ E n . This implies that it can be represen
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  Chapter 2 Projection Matrices 2.1 Definition Definition 2.1 Let  x ∈ E  n = V   ⊕ W  . Then  x can be uniquely decomposed into x = x 1 + x 2 ( where  x 1 ∈ V   and  x 2 ∈ W  ) . The transformation that maps  x into x 1 is called the projection matrix (or simply projector) onto V   along  W  and is denoted as  φ . This is a linear transformation; that is, φ ( a 1 y 1 + a 2 y 2 ) = a 1 φ ( y 1 ) + a 2 φ ( y 2 ) (2.1)  for any  y 1 , y 2 ∈ E  n . This implies that it can be represented by a matrix.This matrix is called a projection matrix and is denoted by  P  V  · W  . The vec-tor transformed by  P  V  · W  (that is, x 1 = P  V  · W  x ) is called the projection (or the projection vector) of  x onto V   along  W  . Theorem 2.1 The necessary and sufficient condition for a square matrix  P  of order  n to be the projection matrix onto V   = Sp( P  ) along  W  = Ker( P  ) is given by  P  2 = P  . (2.2)We need the following lemma to prove the theorem above. Lemma 2.1 Let  P  be a square matrix of order  n , and assume that (2.2)holds. Then  E  n = Sp( P  ) ⊕ Ker( P  ) (2.3) © Springer Science+Business Media, LLC 2011Statistics for Social and Behavioral Sciences, DOI 10.1007/978-1-4419-9887-3_2, 25 H. Yanai et al.,  Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition ,  26 CHAPTER 2. PROJECTION MATRICES  and  Ker( P  ) = Sp( I  n − P  ) . (2.4) Proof of Lemma 2.1. (2.3): Let x ∈ Sp( P  ) and y ∈ Ker( P  ). From x = Pa , we have Px = P  2 a = Pa = x and Py = 0 . Hence, from x + y = 0 ⇒ Px + Py = 0 , we obtain Px = x = 0 ⇒ y = 0 . Thus, Sp( P  ) ∩ Ker( P  ) = { 0 } . On the other hand, from dim(Sp( P  )) + dim(Ker( P  )) =rank( P  ) + ( n − rank( P  )) = n , we have E  n = Sp( P  ) ⊕ Ker( P  ).(2.4): We have Px = 0 ⇒ x = ( I  n − P  ) x ⇒ Ker( P  ) ⊂ Sp( I  n − P  ) onthe one hand and P  ( I  n − P  ) ⇒ Sp( I  n − P  ) ⊂ Ker( P  ) on the other. Thus,Ker( P  ) = Sp( I  n − P  ). Q.E.D. Note When (2.4) holds, P  ( I  n − P  ) = O ⇒ P  2 = P  . Thus, (2.2) is the necessaryand sufficient condition for (2.4). Proof of Theorem 2.1. (Necessity) For ∀ x ∈ E  n , y = Px ∈ V   . Notingthat y = y + 0 , we obtain P  ( Px ) = Py = y = Px = ⇒ P  2 x = Px = ⇒ P  2 = P  . (Sufficiency) Let V   = { y | y = Px , x ∈ E  n } and W  = { y | y = ( I  n − P  ) x , x ∈ E  n } . From Lemma 2.1, V   and W  are disjoint. Then, an arbitrary x ∈ E  n can be uniquely decomposed into x = Px + ( I  n − P  ) x = x 1 + x 2 (where x 1 ∈ V   and x 2 ∈ W  ). From Definition 2.1, P  is the projection matrixonto V   = Sp( P  ) along W  = Ker( P  ). Q.E.D.Let E  n = V   ⊕ W  , and let x = x 1 + x 2 , where x 1 ∈ V   and x 2 ∈ W  . Let P  W  · V  denote the projector that transforms x into x 2 . Then, P  V  · W  x + P  W  · V  x = ( P  V  · W  + P  W  · V  ) x . (2.5)Because the equation above has to hold for any x ∈ E  n , it must hold that I  n = P  V  · W  + P  W  · V  . Let a square matrix P  be the projection matrix onto V   along W  . Then, Q = I  n − P  satisfies Q 2 = ( I  n − P  ) 2 = I  n − 2 P  + P  2 = I  n − P  = Q ,indicating that Q is the projection matrix onto W  along V   . We also have PQ = P  ( I  n − P  ) = P  − P  2 = O , (2.6)  2.1. DEFINITION  27implying that Sp( Q ) constitutes the null space of  P  (i.e., Sp( Q ) = Ker( P  )).Similarly, QP  = O , implying that Sp( P  ) constitutes the null space of  Q (i.e., Sp( P  ) = Ker( Q )). Theorem 2.2 Let  E  n = V   ⊕ W  . The necessary and sufficient conditions  for a square matrix  P  of order  n to be the projection matrix onto V   along  W  are: (i) Px = x for ∀ x ∈ V, (ii) Px = 0 for ∀ x ∈ W. (2.7) Proof. (Sufficiency) Let P  V  · W  and P  W  · V  denote the projection matricesonto V   along W  and onto W  along V   , respectively. Premultiplying (2.5) by P  , we obtain P  ( P  V  · W  x ) = P  V  · W  x , where PP  W  · V  x = 0 because of (i) and(ii) above, and P  V  · W  x ∈ V   and P  W  · V  x ∈ W  . Since Px = P  V  · W  x holdsfor any x , it must hold that P  = P  V  · W  .(Necessity) For any x ∈ V   , we have x = x + 0 . Thus, Px = x . Similarly,for any y ∈ W  , we have y = 0 + y , so that Py = 0 . Q.E.D. Example 2.1 InFigure 2.1, −→ OA indicates the projection of  z onto Sp( x )along Sp( y ) (that is, −→ OA = P  Sp ( x ) · Sp ( y ) z ), where P  Sp ( x ) · Sp ( y ) indicates theprojection matrix onto Sp( x ) along Sp( y ). Clearly, −→ OB = ( I  2 − P  Sp ( y ) · Sp ( x ) ) × z .Sp( y ) = { y } Sp( x ) = { x } AB P  { x }·{ y } z O z Figure 2.1: Projection onto Sp( x ) = { x } along Sp( y ) = { y } . Example 2.2 InFigure 2.2, −→ OA indicates the projection of  z onto V   = { x | x = α 1 x 1 + α 2 x 2 } along Sp( y ) (that is, −→ OA = P  V  · Sp ( y ) z ), where P  V  · Sp ( y ) indicates the projection matrix onto V   along Sp( y ).  28 CHAPTER 2. PROJECTION MATRICES  Sp( y ) = { y } V   = { α 1 x 1 + α 2 x 2 } AB P  V  ·{ y } z O z Figure 2.2: Projection onto a two-dimensional space V   along Sp( y ) = { y } . Theorem 2.3 The necessary and sufficient condition for a square matrix  P  of order  n to be a projector onto V   of dimensionality  r (dim( V   ) = r ) is given by  P  = T  ∆ r T  − 1 , (2.8) where  T  is a square nonsingular matrix of order  n and  ∆ r =  1 ··· 0 0 ··· 0 .................. 0 ··· 1 0 ··· 00 ··· 0 0 ··· 0 .................. 0 ··· 0 0 ··· 0  . (There are  r unities on the leading diagonals, 1 ≤ r ≤ n .) Proof. (Necessity) Let E  n = V   ⊕ W  , and let A = [ a 1 , a 2 , ··· , a r ] and B = [ b 1 , b 2 , ··· b n − r ] be matrices of linearly independent basis vectors span-ning V   and W  , respectively. Let T  = [ A , B ]. Then T  is nonsingular,since rank( A ) + rank( B ) = rank( T  ). Hence, ∀ x ∈ V   and ∀ y ∈ W  can beexpressed as x = Aα = [ A , B ]  α 0  = T   α 0  , y = Aα = [ A , B ]  0 β  = T   0 β  .
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