Exemplo 4.34. Determine o operador de projeção ortogonal sobre o espaço W gerado pelos vetores u=\left[\begin{array}{c}1 \\\\ -1 \\\\ 0\end{array}\right] e v=\left[\begin{array}{l}0 \\\\ 1 \\\\ 2\end{array}\right].
Temos, A=\left[\begin{array}{cc}1 & 0 \\\\ -1 & 1 \\\\ 0 & 2\end{array}\right], \log A^{t} A=\left[\begin{array}{ccc}1 & -1 & 0 \\\\ 0 & 1 & 2\end{array}\right]\left[\begin{array}{cc}1 & 0 \\\\ -1 & 1 \\\\ 0 & 2\end{array}\right]=\left[\begin{array}{cc}2 & -1 \\\\ -1 & 5\end{array}\right] \quad e \left(A^{t} A\right)^{-1}=\frac{1}{9}\left[\begin{array}{ll}5 & 1 \\\\ 1 & 2\end{array}\right], assim,
\left[P_{W}\right]_{C}=A\left(A^{t} A\right)^{-1} A^{t}=\frac{1}{9}\left[\begin{array}{cc} 1 & 0 \\\\ -1 & 1 \\\\ 0 & 2 \end{array}\right]\left[\begin{array}{ll} 5 & 1 \\\\ 1 & 2 \end{array}\right]\left[\begin{array}{ccc} 1 & -1 & 0 \\\\ 0 & 1 & 2 \end{array}\right]=\left[\begin{array}{ccc} 5 & -4 & 2 \\\\ -4 & 5 & 2 \\\\ 2 & 2 & 8 \end{array}\right]
portanto
P_{W}(x, y, z)=\frac{1}{9}\left[\begin{array}{c} 5 x-4 y+2 z \\\\ -4 x+5 y+2 z \\\\ 2 x+2 y+8 z \end{array}\right]

Passo 1

O enunciado pede o operador de projeção ortogonal P_{W} sobre o subespaço W\subset\mathbb{R}^3 gerado pelos vetores
u = \begin{pmatrix}1\\\ -1\\\ 0\end{pmatrix},\quad v = \begin{pmatrix}0\\\ 1\\\ 2\end{pmatrix}.
A teoria de projeção ortogonal diz que, se montarmos a matriz A\in\mathbb{R}^{3\times2} cujas colunas são u e v, então
P_{W} \;=\; A\,(A^{T}A)^{-1}A^{T}.

Passo 2

Montamos
A = \begin{pmatrix} 1 & 0\\\\ -1& 1\\\\ 0 & 2 \end{pmatrix},
e calculamos
A^{T}A = \begin{pmatrix}1 & -1 & 0\\\ 0 & 1 & 2\end{pmatrix} \begin{pmatrix}1 & 0\\\ -1& 1\\\ 0 & 2\end{pmatrix} = \begin{pmatrix}2 & -1\\\ -1 & 5\end{pmatrix}.

Passo 3

Invertendo a matriz A^{T}A, obtemos
(A^{T}A)^{-1} = \frac{1}{9}\begin{pmatrix}5 & 1\\\ 1 & 2\end{pmatrix}.

Passo 4

Agora calculamos
P_{W} = A\,(A^{T}A)^{-1}A^{T} = \frac{1}{9} \begin{pmatrix}1 & 0\\\ -1 & 1\\\ 0 & 2\end{pmatrix} \begin{pmatrix}5 & 1\\\ 1 & 2\end{pmatrix} \begin{pmatrix}1 & -1 & 0\\\ 0 & 1 & 2\end{pmatrix}.
Multiplicando passo a passo chega-se à matriz
[P_{W}] = \frac{1}{9} \begin{pmatrix} 5 & -4 & 2\\\\ -4 & 5 & 2\\\\ 2 & 2 & 8 \end{pmatrix}.

Resposta

Para (x,y,z)\in\mathbb{R}^3, a projeção ortogonal em W é
P_{W}(x,y,z) = \frac{1}{9} \begin{pmatrix} 5x - 4y + 2z\\\\ -4x + 5y + 2z\\\\ 2x + 2y + 8z \end{pmatrix}.

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